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## Solving absolute value equations and inequalities

• Absolute equations I
• Absolute equations II
• Absolute equations III

The absolute number of a number a is written as

$$\left | a \right |$$

And represents the distance between a and 0 on a number line.

An absolute value equation is an equation that contains an absolute value expression. The equation

$$\left | x \right |=a$$

Has two solutions x = a and x = -a because both numbers are at the distance a from 0.

To solve an absolute value equation as

$$\left | x+7 \right |=14$$

You begin by making it into two separate equations and then solving them separately.

$$x+7 =14$$

$$x+7\, {\color{green} {-\, 7}}\, =14\, {\color{green} {-\, 7}}$$

$$x+7 =-14$$

$$x+7\, {\color{green} {-\, 7}}\, =-14\, {\color{green} {-\, 7}}$$

An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative.

The inequality

$$\left | x \right |<2$$

Represents the distance between x and 0 that is less than 2

Whereas the inequality

$$\left | x \right |>2$$

Represents the distance between x and 0 that is greater than 2

You can write an absolute value inequality as a compound inequality.

$$\left | x \right |<2\: or$$-2<x<2$$This holds true for all absolute value inequalities.$$\left | ax+b \right |<c,\: where\: c>0=-c<ax+b<c\left | ax+b \right |>c,\: where\: c>0=ax+b<-c\: or\: ax+b>c$$You can replace > above with ≥ and < with ≤. When solving an absolute value inequality it's necessary to first isolate the absolute value expression on one side of the inequality before solving the inequality. Solve the absolute value inequality$$2\left |3x+9 \right |<36\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}\left | 3x+9 \right |<18-18<3x+9<18-18\, {\color{green} {-\, 9}}<3x+9\, {\color{green} {-\, 9}}<18\, {\color{green} {-\, 9}}-27<3x<9\frac{-27}{{\color{green} 3}}<\frac{3x}{{\color{green} 3}}<\frac{9}{{\color{green} 3}}-9<x<3$$## Video lesson Solve the absolute value equation$$4 \left |2x -1 \right | -2 = 10$$• Graphing linear systems • The substitution method for solving linear systems • The elimination method for solving linear systems • Systems of linear inequalities • Properties of exponents • Scientific notation • Exponential growth functions • Monomials and polynomials • Special products of polynomials • Polynomial equations in factored form • Use graphing to solve quadratic equations • Completing the square • The quadratic formula • The graph of a radical function • Simplify radical expressions • Radical equations • The Pythagorean Theorem • The distance and midpoint formulas • Simplify rational expression • Multiply rational expressions • Division of polynomials • Add and subtract rational expressions • Solving rational equations • Algebra 2 Overview • Geometry Overview • SAT Overview • ACT Overview ## Solving Absolute Value Equations and Inequalities As we saw earlier in the Negative Numbers and Absolute Value section, an absolute value (designated by | |) means take the positive value of whatever is between the two bars. The absolute value is always positive, so you can think of it as the distance from 0 . As an example,  \left| 3 \right|=3 and  \left| {-3} \right|=3. It’s as simple as that! (Note that we also address absolute values here in the Piecewise Functions section and here in the Rational Functions, Equations, and Inequalities section .) ## Solving Absolute Value Equations Solving absolute equations isn’t too difficult; just have to separate the equation into two different equations (once we isolate the absolute value), since we don’t if what’s inside the absolute value is positive or negative . Then, make the expression on the right-hand side (without the variables) both positive and negative and solve each equation; typically, we will get two answers . We must check our answer s , since we may get extraneous solutions (solutions that don’t work). . There are a few cases with absolute value equations or inequalities where we don’t have to even solve! One is when we have isolated the absolute value, and it is set equal to a negative number , such as  \left| {x-5} \right|=-4, or  \left| {x-5} \right|\le -4, for example. Since an absolute value can never be negative , we have no solution for this case. The other is when the absolute value is greater than a negative number, such as  \left| {x-5} \right|>-4 for example. In this case our answer is all real numbers , since an absolute value is always positive. Note that we can always solve absolute value equations and inequalities graphically , as shown below. Here are some problems: Here’s one more that’s a bit tricky, since we have two expressions with absolute value in it. In this case, we have to separate in four cases , just to be sure we cover all the possibilities. We then must check for extraneous solutions , possible solutions that don’t work. For example, when the expression  3x-2 is negative, the absolute value of that expression is the negation of it, or  -3x+2, to make it positive in the equation. Play around with some numbers and you’ll see this! When we get all the possible answers, check for extraneous solutions , since we’re dealing with absolute value. We found two answers that worked:  \displaystyle x=\frac{3}{2} and  x=-1. You can also put the equation in your graphing calculator to check your answers! Here’s another way to approach the absolute value problem above, using number lines : Now draw number lines for each absolute value, and then for the whole equation above . We see for the last number line that for  <-2, we’ll use  2-3x and  -x-2, between  -2 and  \displaystyle \frac{2}{3}, we’ll use  2-3x and  x+2 , and  \displaystyle >\frac{2}{3}, we’ll use  3x-2 and  x+2. After solving for  x in the original equation, we have to check to make sure each value we get for  x falls into the correct interval of the number line. For example,  \displaystyle -\frac{3}{2} isn’t  <-2, so we have to “throw it away”.  -1 is between  -2 and  \displaystyle \frac{2}{3}, so it works, and  \displaystyle \frac{3}{2} is  \displaystyle >\frac{2}{3}, so it works. We have  \displaystyle x=\frac{3}{2} and  x=-1. √ ## Solving Absolute Value Inequalities Note that we learned about Linear Inequalities here . When dealing with absolute values and inequalities (just like with absolute value equations), we have to separate the inequality into two different ones , if there are any variables inside the absolute value bars. First, get the absolute value all by itself on the left (remember to reverse the inequality sign when multiplying or dividing by a negative number). Now, separate the equations. We get the first equation by just taking away the absolute value sign away on the left. The easiest way to get the second equation is to take the absolute value sign away on the left, and do two things on the right : reverse the inequality sign , and change the sign of everything on the right (even if we have variables over there). We also have to think about whether or not to use “ or ” or “ and ” between the two new equations. The way I remember this is that with a  >\,\text{or}\,\,\ge  sign, you can remember “gore”: greater than uses “or” . With a  <\,\text{or}\,\,\le  sign, think “land”: less than uses “and” . GORE: Greater Than ­­­uses OR LAND: Less Than uses AND Note that statement with “or” is a disjunction , which means that it works if only one (or both) parts are true. A statement with “and” is a conjunction , which means it only works if both parts are true. And again, if we get something like  \left| {x+3} \right|<0 (or a negative number), there is no solution , and something like  \left| {x+3} \right|\ge 0 (or a negative number), there are infinite solutions (all real numbers). Also, remember to use open brackets for inequalities that aren’t inclusive ( < and  <) and closed brackets for inequalities that are inclusive and include the boundary point ( \le  and  \ge ). Here are some examples: There are examples of rational functions with absolute values here in the Rational Functions, Equations, and Inequalities section . ## Graphs of Absolute Value Functions Note that you can put absolute values in your Graphing Calculator (and even graph them!) by hitting MATH, scroll right to NUM , and then hitting 1 (abs) or ENTER . Absolute Value functions typically look like a V (upside down if the absolute value is negative), where the point at the V is called the vertex . For the absolute value parent function, the vertex is at  \left( {0,0} \right). We looked at absolute value parent functions and their transformations in the Absolute Value Transformations section , and absolute value functions as piecewise equations here in the Piecewise Functions section . Note that the general form for the absolute value function is  f\left( x \right)=a\left| {x-h} \right|+k, where  \left( {h,k} \right) is the vertex. If  a is positive, the function points down (like a V ); if  a is negative, the function points up (like an upside-down V ). Here’s a graph of the parent function, and also a transformation: Without using a t-chart, we can see that the vertex is at  \left( {-2,1} \right) and the graph is upside-down because of the negative sign. It’s also stretched vertically by a factor of 3 and horizontal by a factor of  \displaystyle \frac{1}{2} (or stretched vertically by a factor of 6 ); thus, other points down can be drawn by going back and forth 1 and down 6 . You can solve absolute value equations and equalities with graphing ; here are some examples of solving inequalities: ## Applications of Absolute Value Functions Absolute Value Functions are in many applications , especially in those involving V-shaped paths and margin of errors , or tolerances . Here are some examples absolute value “word” problems that you may see: Here are examples that are absolute value inequality applications . Use this rule of thumb : the absolute value of a difference is usually on the left-hand side, the amount that differs or varies is usually on the right-hand side, with a  < or  \le  sign in between. Learn these rules, and practice, practice, practice! Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems. If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software. You can even get math worksheets. You can also go to the Mathway site here , where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy! On to Solving Radical Equations and Inequalities – you’re ready! If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. ## Unit 1: Linear equations and inequalities About this unit. Linear equations and inequalities are the foundation of many advanced math topics, such as functions, systems, matrices, and calculus. Learn how to master them and unlock new possibilities for your future studies and careers in engineering, finance, computer science, and more. ## Solving equations with one unknown • Equations with parentheses (Opens a modal) • Multi-step equations review (Opens a modal) • Number of solutions to equations (Opens a modal) • Worked example: number of solutions to equations (Opens a modal) • Equations with parentheses Get 3 of 4 questions to level up! • Equations with parentheses: decimals & fractions Get 3 of 4 questions to level up! • Number of solutions to equations Get 3 of 4 questions to level up! ## Solutions to linear equations • Intro to the coordinate plane (Opens a modal) • Solutions to 2-variable equations (Opens a modal) • Worked example: solutions to 2-variable equations (Opens a modal) • Creativity break: Why is creativity important in algebra? (Opens a modal) • Interpreting points in context of graphs of systems (Opens a modal) • Solutions to 2-variable equations Get 3 of 4 questions to level up! • Interpret points relative to a system Get 3 of 4 questions to level up! ## Multi-step linear inequalities • Inequalities with variables on both sides (Opens a modal) • Inequalities with variables on both sides (with parentheses) (Opens a modal) • Multi-step inequalities (Opens a modal) • Multi-step linear inequalities Get 3 of 4 questions to level up! ## Compound inequalities • Compound inequalities: OR (Opens a modal) • Compound inequalities: AND (Opens a modal) • A compound inequality with no solution (Opens a modal) • Double inequalities (Opens a modal) • Compound inequalities review (Opens a modal) • Compound inequalities Get 3 of 4 questions to level up! ## Modeling with linear equations and inequalities • Comparing linear rates example (Opens a modal) • Comparing linear rates word problems Get 3 of 4 questions to level up! ## Absolute value equations • Intro to absolute value equations and graphs (Opens a modal) • Solving absolute value equations (Opens a modal) • Absolute value equations Get 3 of 4 questions to level up! ## 1-5 Absolute Value Equations and Inequalities Here is your free content for this lesson! ## Absolute Value Equations and Inequalities - Word Docs & PowerPoints To gain access to our editable content Join the Algebra 2 Teacher Community! Here you will find hundreds of lessons, a community of teachers for support, and materials that are always up to date with the latest standards. ## Absolute Value Equations and Inequalities Worksheet, Notes - PDFs 1-5 Assignment - Absolute Value Equations and Inequalities 1-5 Assignment SE - Absolute Value Equations and Inequalities 1-5 Bell Work - Absolute Value Equations and Inequalities 1-5 Bell Work SE - Absolute Value Equations and Inequalities 1-5 Exit Quiz - Absolute Value Equations and Inequalities 1-5 Exit Quiz SE - Absolute Value Equations and Inequalities 1-5 Guided Notes SE - Absolute Value Equations and Inequalities 1-5 Guided Notes TE - Absolute Value Equations and Inequalities 1-5 Lesson Plan - Absolute Value Equations and Inequalities 1-5 Online Activity - Absolute Value Equations and Inequalities 1-5 Slide Show - Absolute Value Equations and Inequalities ## Matrix Multiplication The properties of parabolas, organizing data into matrices, modeling data with quadratic functions, teaching linear equations, graphing systems of equations, teaching relations and functions, solving systems of linear inequalities, solving equations (algebra 2), simplifying algebraic expressions (algebra 2), angles and the unit circle – time to eat, properties of real numbers – the importance of differentiating directions in algebra, absolute value functions and graphs – real world applications, rational functions and their graphs – group activity, algebraic expressions worksheet and activity – mazing, holiday algebra 2 activities, how to make your math class paperless, i stopped letting my students use calculators in class, algebra 2 teacher hacks, conditional probability – call it in the air, area under a curve – is your umbrella big enough, properties of logarithms, adding and subtracting matrices – using rainbows, the nightmare of exploring conic sections, probability of multiple events – a coin and a card, roots and radical expressions – why so negative, solving systems of equations by substitution – sports and algebra 2, permutations and combinations using magic card tricks, angry birds parabola project, share this:. • Click to share on Facebook (Opens in new window) • Click to share on Pinterest (Opens in new window) • Click to share on Twitter (Opens in new window) • Click to share on Reddit (Opens in new window) • Click to email this to a friend (Opens in new window) • Click to print (Opens in new window) • Click to share on LinkedIn (Opens in new window) • Click to share on Pocket (Opens in new window) • Click to share on Tumblr (Opens in new window) • Click to share on WhatsApp (Opens in new window) • Click to share on Skype (Opens in new window) • Click to share on Telegram (Opens in new window) • school Campus Bookshelves • menu_book Bookshelves • perm_media Learning Objects • login Login • how_to_reg Request Instructor Account • hub Instructor Commons ## Margin Size • Download Page (PDF) • Download Full Book (PDF) • Periodic Table • Physics Constants • Scientific Calculator • Reference & Cite • Tools expand_more • Readability selected template will load here This action is not available. ## 1.4: Absolute value inequalities • Last updated • Save as PDF • Page ID 48950 • Thomas Tradler and Holly Carley • CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ ( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$ $$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$ $$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$ $$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vectorC}[1]{\textbf{#1}}$$ $$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$ $$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$ $$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$ Using the notation from the previous section, we now solve inequalities involving the absolute value. These inequalities may be solved in three steps: • Step 1: Solve the corresponding equality . The solution of the equality divides the real number line into several subintervals. • Step 2: Using step 1, check the inequality for a number in each of the subintervals. This check determines the intervals of the solution set. • Step 3: Check the endpoints of the intervals. Here are some examples for the above solution method. ## Example $$\PageIndex{1}$$ Solve for $$x$$ : • $$|x+7|<2$$ • $$|3x-5|\geq 11$$ • $$|12-5x|\leq 1$$ • We follow the three steps described above. In step 1, we solve the corresponding equality, $$|x+7|=2$$ . $$x+7=2$$ & $$x+7=-2$$ $\begin{array}{l|l} x+7=2 & x+7=-2 \\ \Longrightarrow x=-5 & \Longrightarrow x=-9 \end{array} \nonumber$ The solutions $$x=-5$$ and $$x=-9$$ divide the number line into three subintervals: Now, in step 2, we check the inequality for one number in each of these subintervals. $\begin{array}{c|c|c} \text { Check: } \quad x=-10 & \text { Check: } \quad x=-7 & \text { Check: } \quad x=0 \\ |(-10)+7| \stackrel{?}{<} 2 & |(-7)+7| \stackrel{?}{<} 2 & |0+7| \stackrel{?}{<} 2 \\ |-3| \stackrel{?}{<} 2 & |0| \stackrel{?}{<} 2 & |7| \stackrel{?}{<} 2 \\ 3 \stackrel{?}{<} 2 & 0 \stackrel{?}{<} 2 & 7 \stackrel{?}{<} 2 \\ \text { false } & \text { true } & \text { false } \end{array} \nonumber$ Since $$x=-7$$ in the subinterval given by $$-9<x<-5$$ solves the inequality $$|x+7|<2$$ , it follows that all numbers in the subinterval given by $$-9<x<-5$$ solve the inequality. Similarly, since $$x=-10$$ and $$x=0$$ do not solve the inequality, no number in these subintervals will solve the inequality. For step 3, we note that the numbers $$x=-9$$ and $$x=-5$$ are not included as solutions since the inequality is strict (that is we have $$<$$ instead of $$\leq$$ ).The solution set is therefore the interval $$S=(-9,-5)$$ . The solution on the number line is: • We follow the steps as before. First, in step 1, we solve $$|3x-5|=11$$ . $\begin{array}{l|l} 3 x-5=11 & 3 x-5=-11 \\ \Longrightarrow 3 x=16 & \Longrightarrow 3 x=-6 \\ \Longrightarrow x=\dfrac{16}{3} & \Longrightarrow x=-2 \end{array} \nonumber$ The two solutions $$x=-2$$ and $$x=\dfrac{16}{3}=5\dfrac {1}{3}$$ divide the number line into the subintervals displayed below. $x<-2 \hspace{1in} -2<x<5\frac 1 3 \hspace{1in} 5\frac 1 3<x \nonumber$ For step 2, we check a number in each subinterval. This gives: $\begin{array}{c|c|c|c} \text { Check: } x=-3 & \text { Check: } \quad x=1 & \text { Check: } \quad x=6 \\ |3 \cdot(-3)-5| \stackrel{?}{\geq} 11 & |3 \cdot 1-5| \stackrel{?}{\geq} 11 & |3 \cdot 6-5| \stackrel{?}{\geq} 11 \\ |-9-5| \stackrel{?}{\geq} 11 & |3-5| \stackrel{?}{\geq} 11 & |18-5| \stackrel{?}{\geq} 11 \\ |-14| \stackrel{?}{\geq} 11 & |-2| \stackrel{?}{\geq} 11 & |13| \stackrel{?}{\geq} 11 \\ 14 \stackrel{?}{\geq} 11 & 2 \stackrel{?}{\geq} 11 & 13 \stackrel{?}{\geq} 11 \\ \text { true } & \text { false } & \text { true } \end{array} \nonumber$ For step 3, note that we include $$-2$$ and $$5\dfrac {1}{3}$$ in the solution set since the inequality is “greater than or equal to” (that is $$\geq$$ , as opposed to $$>$$ ). Furthermore, the numbers $$-\infty$$ and $$\infty$$ are not included, since $$\pm\infty$$ are not real numbers. The solution set is therefore the union of the two intervals: $S=\Big(-\infty,-2\Big]\cup \Big[5\dfrac {1}{3}, \infty\Big) \nonumber$ • To solve $$|12-5x|\leq 1$$ , we first solve the equality $$|12-5x|=1$$ . $\begin{array}{l|l} 12-5 x=1 & 12-5 x=-1 \\ \Longrightarrow-5 x=-11 & \Longrightarrow-5 x=-13 \\ \Longrightarrow x=\frac{-11}{-5}=2.2 & \Longrightarrow x=\frac{-13}{-5}=2.6 \end{array} \nonumber$ This divides the number line into three subintervals, and we check the original inequality $$|12-5x|\leq 1$$ for a number in each of these subintervals. $\begin{array}{c|c|c|c} \text {Interval: } \quad x<2.2 & \text {Interval: } \quad 2.2<x<2.6 & \text {Interval: } \quad 2.6<x\\ \text {Check: } \quad x=1 & \text {Check: } \quad x=2.4 & \text {Check: } \quad x=3 \\ |12-5 \cdot 1| \stackrel{?}{\leq} 1 & |12-5 \cdot 1| \stackrel{?}{\leq} 1 & |12-5 \cdot 3| \stackrel{?}{\leq} 1 \\ |12-5| \stackrel{?}{\leq} 1 & |12-12| \stackrel{?}{\leq} 1 & |12-15| \stackrel{?}{\leq} 1 \\ |7| \stackrel{?}{\leq} 1 & |0| \stackrel{?}{\leq} 1 & |-3| \stackrel{?}{\leq} 1 \\ 7 \stackrel{?}{\leq} 1 & 0 \stackrel{?}{\leq} 1 & 3 \stackrel{?}{\leq} 1 \\ \text { false } & \text { true } & \text { false } \end{array} \nonumber$ The solution set is the interval $$S=[2.2,2.6]$$ , where we included $$x=2.2$$ and $$x=2.6$$ since the original inequality “less than or equal to” ( $$\leq$$ ) includes the equality. Alternatively, whenever you have an absolute value inequality you can turn it into two inequalities. Here are a couple of examples. ## Example $$\PageIndex{2}$$ Solve for $$x$$ : $$|12-5x|\leq 1$$ Note that $$|12-5x|\leq 1$$ implies that $-1\leq 12-5x\leq1 \nonumber$ $-13\leq -5x\leq -11 \nonumber$ and by dividing by $$-5$$ (remembering to switch the direction of the inequalities when multiplying or dividing by a negative number) we see that $\frac{13}{5}\geq x\geq \frac{11}{5} \nonumber$ or in interval notation, we have the solution set $S=\left[\frac{11}{5},\frac{13}{5}\right] \nonumber$ ## Example $$\PageIndex{3}$$ If $$|x+6|>2$$ then either $$x+6>2$$ or $$x+6<-2$$ so that either $$x>-4$$ or $$x<-8$$ so that in interval notation the solution is $$S=(-\infty,-8)\cup(-4,\infty)$$. There is a geometric interpretation of the absolute value on the number line as the distance between two numbers: distance between $$a$$ and $$b$$ is $$|b-a|$$ which is also equal to $$|a-b|$$ This interpretation can also be used to solve absolute value equations and inequalities. ## Example $$\PageIndex{4}$$ • $$|x-6|=4$$ • $$|x-6|\leq 4$$ • $$|x-6|\geq 4$$ • Consider the distance between $$x$$ and $$6$$ to be $$4$$ on a number line: There are two solutions, $$x=2$$ or $$x=10$$ . That is, the distance between $$2$$ and $$6$$ is $$4$$ and the distance between $$10$$ and $$6$$ is $$4$$ . • Numbers inside the braces above have distance $$4$$ or less. The solution is given on the number line as: In interval notation, the solution set is the interval $$S=[2,10]$$ . One can also write that the solution set consists of all $$x$$ such that $$2\leq x\leq 10$$ . • Numbers outside the braces above have distance $$4$$ or more. The solution is given on the number line as: In interval notation, the solution set is the interval $$(-\infty,2]$$ and $$[10,\infty)$$ , or in short it is the union of the two intervals: $S= (\infty,2]\cup [10,\infty) \nonumber$ One can also write that the solution set consists of all $$x$$ such that $$x\leq 2$$ or $$x\geq 10$$ . • For Parents • For Teachers • Teaching Topics • Kindergarten • EM3/CCSS at Home • Family Letters • Student Gallery • Understanding EM • Algorithms/ Computation • Student Links ## EM4 at Home Expressions and equations. Everyday Mathematics for Parents: What You Need to Know to Help Your Child Succeed The University of Chicago School Mathematics Project University of Chicago Press Learn more >> ## Related Links Help with algorithms. Access video tutorials, practice exercises, and information on the research basis and development of various algorithms. ## Everyday Mathematics Online With a login provided by your child's teacher, access resources to help your child with homework or brush up on your math skills. ## Parent Connections on Publisher's site McGraw-Hill Education offers many resources for parents, including tips, activities, and helpful links. ## Parent Resources on EverydayMath.com EverydayMath.com features activity ideas, literature lists, and family resources for the EM curriculum. ## Understanding Everyday Mathematics for Parents Learn more about the EM curriculum and how to assist your child. #### IMAGES 1. Absolute Value Equations and Inequalities Activity FREE: First Step, Stop! 2. Absolute Value Inequalities Worksheet Answers 3. Unit 1: Expressions, Equations and Inequalities 4. Absolute Value Equations And Inequalities Worksheet 5. Equations and Inequalities (Algebra 2 Curriculum Unit 1) 6. Absolute Value Inequalities Worksheet Answers Algebra 1 #### VIDEO 1. Solving Absolute Value Equations and Inequality Part 1 / Sec 2 2. Absolute Value Equations and Inequalities 3. Section 1.4 4. Maths 9 unit 7 Ex 7.2 Q2 part ( 4 ) #linear equations and inequalities 5. Kuta Software 6. Inequalities: Lesson 4 (absolute value inequalities) #### COMMENTS 1. PDF Algebra 3-4 Unit 1 Absolute Value Functions and Equations 1.2 I can identify increasing, decreasing, and the average rate of change of a given or table of values. 1.2 I can find a linear regression line and use it to predict values. 1.3 I can graph absolute value equations, identifying transformations. 1.4 I can identify min, max, vertex, end behavior, and compare absolute value graphs and tables. 1.5 ... 2. 2.6: Solving Absolute Value Equations and Inequalities Step 2: Set the argument of the absolute value equal to ± p. Here the argument is 5x − 1 and p = 6. 5x − 1 = − 6 or 5x − 1 = 6. Step 3: Solve each of the resulting linear equations. 5x − 1 = − 6 or 5x − 1 = 6 5x = − 5 5x = 7 x = − 1 x = 7 5. Step 4: Verify the solutions in the original equation. Check x = − 1. 3. Unit 8: Absolute value equations, functions, & inequalities About this unit. This topic covers: Solving absolute value equations. Graphing absolute value functions. Solving absolute value inequalities. 4. Absolute Value Equations and Inequalities Flashcards Quiz: solving absolute value equations and inequalities 1.4.3. 10 terms. demiann04. Preview. Math Chapter 4 vocabulary. 15 terms. peyton_vance76. Preview. Solving Absolute Value Equations. 10 terms. Carliana2. ... Can be anything except for the number that makes the absolute value expression equal zero. (-∞,-4)U(-4,∞) IxI≥0 , IxI>-3, or ... 5. Algebra II: Absolute Value Equations and Inequalities Quiz Part 1 D. x = 5/2 or x = 1. Solve the inequality. Graph the solution. 8|x + 3/4| < 2. C. - 1 < x < -1/2. A furniture maker uses the specification 19.88 ≤ w ≤ 20.12 for the width w in inches of a desk drawer. Write the specification as an absolute value inequality. B. |w - 20| ≤ 0.12. Study with Quizlet and memorize flashcards containing terms ... 6. Solving Absolute Value Equations and Inequalities This algebra video tutorial shows you how to solve absolute value equations with inequalities and how to plot the solution on a number line and write the ans... 7. Solving Absolute-Value Equations & Inequalities, Part 2 Review how to solve equations and inequalities involving absolute value with these interactive flashcards. 8. Solving absolute value equations and inequalities The absolute number of a number a is written as$$\left | a \right |$$And represents the distance between a and 0 on a number line. An absolute value equation is an equation that contains an absolute value expression. The equation$$\left | x \right |=a Has two solutions x = a and x = -a because both numbers are at the distance a from 0.

9. 4.4: Absolute Value Inequalities

This is the situation shown in Figure 4.4.1 4.4. 1 (a). The graph of y = |x| is therefore never below the graph of y = −5. Thus, the inequality |x| < −5 has no solution. An alternate approach is to consider the fact that the absolute value of x is always nonnegative and can never be less than −5.

10. 1.7: Solve Absolute Value Inequalities

The absolute value of a number is its distance from zero on the number line. We started with the inequality | x | ≤ 5. We saw that the numbers whose distance is less than or equal to five from zero on the number line were − 5 and 5 and all the numbers between − 5 and 5 (Figure 1.7.4 ). Figure 1.7.4.

11. Solving Absolute Value Equations and Inequalities

Separate into four cases, since we don't know whether $3x-2$ and $x+2$ are positive or negative. Since they are absolute values in the equations, they could be either, but still come out positive. For example, when the expression $3x-2$ is negative, the absolute value of that expression is the negation of it, or $-3x+2$, to make it positive in the equation. Play around with some numbers ...

12. 3.1 Absolute Value Inequality

Section 3.1 Absolute Value Inequalities. A2.5.4 Solve equations and inequalities involving absolute values of linear expressions; Need a tutor? Click this link and get your first session free! Packet. a2_3.1_packet.pdf: File Size: 599 kb: File Type: pdf: Download File.

13. Unit 4; Lesson 1; Solving Absolute Value Equations and Inequalities

Objective: In this lesson, you will solve absolute value linear equations and inequalities and use graphs to verify and represent solutions. ... Unit 4; Lesson 1; Solving Absolute Value Equations and Inequalities. Flashcards. Learn. Test. Match. absolute value. Click the card to flip 👆 ...

14. Equations and Inequalities (Algebra 2 Curriculum Unit 1)

This Equations and Inequalities Unit Bundle includes guided notes, homework assignments, two quizzes, a study guide and a unit test that cover the following topics: • Simplifying Radicals. • Classifying Numbers (the Real Number System) • Order of Operations (Includes absolute value and square roots) • Evaluating Expressions (Includes ...

15. Absolute value & piecewise functions

Absolute value graphs make a V shape, but why do they do that? ... Solving equations & inequalities. Unit 3. Working with units. Unit 4. Linear equations & graphs. Unit 5. Forms of linear equations. ... Unit 10: Absolute value & piecewise functions. 600 possible mastery points. Mastered. Proficient. Familiar. Attempted.

16. Unit 1: Linear equations and inequalities

Linear equations and inequalities: Unit test; About this unit. Linear equations and inequalities are the foundation of many advanced math topics, such as functions, systems, matrices, and calculus. ... Absolute value equations Get 3 of 4 questions to level up! Quiz 2. Level up on the above skills and collect up to 320 Mastery points Start quiz ...

17. 1.8: Linear Inequalities and Absolute Value Inequalities

This page titled 1.8: Linear Inequalities and Absolute Value Inequalities is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

18. Algebra 2

Each lesson has notes/class examples and homework examples. There is plenty of room for the student to show their work and write their answers in the space provided. An answer key is included. Download my FREE sample lessons: Algebra 2 - Unit 1 - Lesson 7 - Solving Absolute Value Equations & Inequalities Algebra 2 - Unit 10 - Lesson 1 ...

19. Absolute Value Equations and Inequalities (Algebra 2

Products. $428.00$675.44 Save \$247.44. View Bundle. Equations and Inequalities MEGA Bundle (Algebra 2 - Unit 1) This is a MEGA Bundle of GUIDED NOTES WITH VIDEO LESSONS for your distance learning needs, homework, daily warm-up, content quizzes, mid-unit and end-unit assessments, review assignments, and cooperative activities for Algebra 2 ...

20. Math 3 Unit 1 Absolute Value Equations and Inequalities

Math 3 Unit 1 Absolute Value Equations and Inequalities. 5.0 (1 review) Absolute value. Click the card to flip 👆. the distance from zero on a number line. Click the card to flip 👆. 1 / 22.

21. 1-5 Absolute Value Equations and Inequalities

Here is your free content for this lesson! Absolute Value Equations and Inequalities - Word Docs & PowerPoints. To gain access to our editable content Join the Algebra 2 Teacher Community! Here you will find hundreds of lessons, a community of teachers for support, and materials that are always up to date with the latest standards.

22. 1.4: Absolute value inequalities

Step 1: Solve the corresponding equality. The solution of the equality divides the real number line into several subintervals. Step 2: Using step 1, check the inequality for a number in each of the subintervals. This check determines the intervals of the solution set. Step 3: Check the endpoints of the intervals.

23. Everyday Mathematics

Finding and Graphing Solution Sets of Inequalities. Home Link 4-10 English Español Selected Answers. ... 4-13. Absolute Value. Home Link 4-13 English Español Selected Answers. ... 4-15. Unit 4 Progress Check. Home Link 4-15 English Español Everyday Mathematics for Parents: What You Need to Know to Help Your Child Succeed.

24. PDF MATH 1215X/1215Y/1215Z: Intermediate Algebra Parts I, II, and III

Homework is also turned in through Remind. MATH 1215X COURSE DESCRIPTION: This 1-credit-hour course includes the first third of an Intermediate Algebra course, including problems in ratio and proportion, unit conversions, solving linear equations, and problems modeled by these, finding equations for lines and