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Equations and Inequalities Involving Signed Numbers

In chapter 2 we established rules for solving equations using the numbers of arithmetic. Now that we have learned the operations on signed numbers, we will use those same rules to solve equations that involve negative numbers. We will also study techniques for solving and graphing inequalities having one unknown.

SOLVING EQUATIONS INVOLVING SIGNED NUMBERS

Upon completing this section you should be able to solve equations involving signed numbers.

Example 1 Solve for x and check: x + 5 = 3

Using the same procedures learned in chapter 2, we subtract 5 from each side of the equation obtaining

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Example 2 Solve for x and check: - 3x = 12

Dividing each side by -3, we obtain

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LITERAL EQUATIONS

  • Identify a literal equation.
  • Apply previously learned rules to solve literal equations.

An equation having more than one letter is sometimes called a literal equation . It is occasionally necessary to solve such an equation for one of the letters in terms of the others. The step-by-step procedure discussed and used in chapter 2 is still valid after any grouping symbols are removed.

Example 1 Solve for c: 3(x + c) - 4y = 2x - 5c

First remove parentheses.

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At this point we note that since we are solving for c, we want to obtain c on one side and all other terms on the other side of the equation. Thus we obtain

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Sometimes the form of an answer can be changed. In this example we could multiply both numerator and denominator of the answer by (- l) (this does not change the value of the answer) and obtain

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The advantage of this last expression over the first is that there are not so many negative signs in the answer.

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The most commonly used literal expressions are formulas from geometry, physics, business, electronics, and so forth.

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Notice in this example that r was left on the right side and thus the computation was simpler. We can rewrite the answer another way if we wish.

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GRAPHING INEQUALITIES

  • Use the inequality symbol to represent the relative positions of two numbers on the number line.
  • Graph inequalities on the number line.

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The symbols are inequality symbols or order relations and are used to show the relative sizes of the values of two numbers. We usually read the symbol as "greater than." For instance, a > b is read as "a is greater than b." Notice that we have stated that we usually read a < b as a is less than b. But this is only because we read from left to right. In other words, "a is less than b" is the same as saying "b is greater than a." Actually then, we have one symbol that is written two ways only for convenience of reading. One way to remember the meaning of the symbol is that the pointed end is toward the lesser of the two numbers.

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In simpler words this definition states that a is less than b if we must add something to a to get b. Of course, the "something" must be positive.

If you think of the number line, you know that adding a positive number is equivalent to moving to the right on the number line. This gives rise to the following alternative definition, which may be easier to visualize.

Example 1 3 < 6, because 3 is to the left of 6 on the number line.

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Example 2 - 4 < 0, because -4 is to the left of 0 on the number line.

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Example 3 4 > - 2, because 4 is to the right of -2 on the number line.

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Example 4 - 6 < - 2, because -6 is to the left of -2 on the number line.

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The mathematical statement x < 3, read as "x is less than 3," indicates that the variable x can be any number less than (or to the left of) 3. Remember, we are considering the real numbers and not just integers, so do not think of the values of x for x < 3 as only 2, 1,0, - 1, and so on.

As a matter of fact, to name the number x that is the largest number less than 3 is an impossible task. It can be indicated on the number line, however. To do this we need a symbol to represent the meaning of a statement such as x < 3.

The symbols ( and ) used on the number line indicate that the endpoint is not included in the set.

Example 5 Graph x < 3 on the number line.

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Note that the graph has an arrow indicating that the line continues without end to the left.

Example 6 Graph x > 4 on the number line.

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Example 7 Graph x > -5 on the number line.

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Example 8 Make a number line graph showing that x > - 1 and x < 5. (The word "and" means that both conditions must apply.)

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Example 9 Graph - 3 < x < 3.

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Example 10 x >; 4 indicates the number 4 and all real numbers to the right of 4 on the number line.

The symbols [ and ] used on the number line indicate that the endpoint is included in the set.

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Example 13 Write an algebraic statement represented by the following graph.

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Example 14 Write an algebraic statement for the following graph.

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Example 15 Write an algebraic statement for the following graph.

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SOLVING INEQUALITIES

Upon completing this section you should be able to solve inequalities involving one unknown.

The solutions for inequalities generally involve the same basic rules as equations. There is one exception, which we will soon discover. The first rule, however, is similar to that used in solving equations.

If the same quantity is added to each side of an inequality , the results are unequal in the same order.

Example 1 If 5 < 8, then 5 + 2 < 8 + 2.

Example 2 If 7 < 10, then 7 - 3 < 10 - 3.

We can use this rule to solve certain inequalities.

Example 3 Solve for x: x + 6 < 10

If we add -6 to each side, we obtain

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Graphing this solution on the number line, we have

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We will now use the addition rule to illustrate an important concept concerning multiplication or division of inequalities.

Suppose x > a.

Now add - x to both sides by the addition rule.

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Now add -a to both sides.

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The last statement, - a > -x, can be rewritten as - x < -a. Therefore we can say, "If x > a, then - x < -a. This translates into the following rule:

If an inequality is multiplied or divided by a negative number, the results will be unequal in the opposite order.

Example 5 Solve for x and graph the solution: -2x>6

To obtain x on the left side we must divide each term by - 2. Notice that since we are dividing by a negative number, we must change the direction of the inequality.

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Take special note of this fact. Each time you divide or multiply by a negative number, you must change the direction of the inequality symbol. This is the only difference between solving equations and solving inequalities.

Once we have removed parentheses and have only individual terms in an expression, the procedure for finding a solution is almost like that in chapter 2.

Let us now review the step-by-step method from chapter 2 and note the difference when solving inequalities.

First Eliminate fractions by multiplying all terms by the least common denominator of all fractions. (No change when we are multiplying by a positive number.) Second Simplify by combining like terms on each side of the inequality. (No change) Third Add or subtract quantities to obtain the unknown on one side and the numbers on the other. (No change) Fourth Divide each term of the inequality by the coefficient of the unknown. If the coefficient is positive, the inequality will remain the same. If the coefficient is negative, the inequality will be reversed. (This is the important difference between equations and inequalities.)

solve step by step math problems

  • A literal equation is an equation involving more than one letter.
  • The symbols are inequality symbols or order relations .
  • a a is to the left of b on the real number line.
  • To solve a literal equation for one letter in terms of the others follow the same steps as in chapter 2.
  • To solve an inequality use the following steps: Step 1 Eliminate fractions by multiplying all terms by the least common denominator of all fractions. Step 2 Simplify by combining like terms on each side of the inequality. Step 3 Add or subtract quantities to obtain the unknown on one side and the numbers on the other. Step 4 Divide each term of the inequality by the coefficient of the unknown. If the coefficient is positive, the inequality will remain the same. If the coefficient is negative, the inequality will be reversed. Step 5 Check your answer.

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Solution - Cumulative probability in the standard normal distribution

Other Ways to Solve

Step-by-step explanation

1. find the cumulative probability of the z-scores values up to 25.

More than 99.9 % of the time, data with a standard normal distribution lies within plus or minus 3.9 standard deviations of the mean. The cumulative probability of the values up to 25 is 1 . p ( x < 25 ) = 1 The cumulative probability that x < 25 is 100 %

2. Find the cumulative probability for the z-scores values greater than 25

The cumulative probability of the values greater than 25 is 0 . 25)=0"> p ( x > 25 ) = 0 The cumulative probability of 25"> x > 25 is 0 %

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Why learn this

The normal distribution is important because we see it often in nature. Suppose we gather many unrelated measures, like human heights, blood pressure readings, or IQ scores. They will follow the normal distribution. We see many normally distributed variables in psychology. For example, reading ability, introversion or job satisfaction. In investing, the normal distribution shows asset class returns. Although these distributions are only roughly normal, they are pretty close, and we can treat them as normal. The normal distribution is easy to work with. Many statistical tests rely on it. Moreover, these tests work well even when the distribution is only approximately normal. For example, if a set's mean and standard deviation are known, and the set follows the normal distribution, we can easily convert between percentiles and raw scores. Any normal distribution can be standardized to a standard normal distribution. That way, we can compare two or more separate data sets. Using standard normal distribution, we can estimate probabilities of events involving normal distribution. This way, we can estimate how tall a person is likely to grow, for instance.

Terms and topics

  • Normal and standard normal distributions

Related links

  • Normal Distribution - Math is Fun
  • Standard Normal Distribution Tables, Z Scores, Probability and Empirical Rule - The Organic Chemistry Tutor
  • Z-table - Statistics by Jim

Latest Related Drills Solved

  • p ( z < 62 )
  • p ( z < 0.703 )
  • p ( 58 < x < 66 )
  • p ( − 2.25 < z < 0 )
  • 0.36)"> 0.36)"> p ( z > 0.36 )
  • 2.92)"> 2.92)"> p ( z > 2.92 )
  • 1.974)"> 1.974)"> p ( z > 1.974 )
  • p ( − 1.42 < z < 2.35 )
  • p ( z < − 2.42 )

Biden has a huge decision to make — one of the most important of his presidency

  • President Biden is a lifelong public servant, and he's behind in polls for reasons he can't change.
  • So, for the sake of the country, Biden has a critical decision to make:
  • Whether or not to step aside so the Democrats can nominate another candidate.

Insider Today

President Joe Biden has dedicated almost his entire professional life to serving our country.

This summer, he will have another critical opportunity to do that.

Specifically, he will have to make a decision that, in my opinion, could determine the future of American democracy.

That decision is this:

Whether to accept his party's nomination to run for a second term… or step aside and let the Democrats nominate another candidate.

Whether or not you believe that "democracy is on the ballot" this fall (or care), you will probably agree that the stakes in this election are unusually high. So, at their convention this summer, the Democrats will want to nominate the  strongest possible candidate .

Based on recent polls, it does not seem that Biden is necessarily the  strongest possible candidate .

And even once he finishes sweeping the primaries, Biden does not necessarily  have to be the candidate.

Because, as political experts explain , Biden could still step aside before the Democratic convention this August and allow the Democrats to nominate someone else.

So, over the next six months, for the sake of his country — or at the very least the Democratic party and its policy objectives — Biden needs to dispassionately assess whether he is, in fact, the strongest possible candidate . If not, he should step aside.

In making this decision, Biden needs to look and think carefully about not what is "true" or "right" or about what he wants most Americans to think, but what Americans actually think.

According to recent polls, even in our polarized country, most Americans actually agree about several things.

One is that ex-President Trump would do a better job than Biden  on the economy (mostly, it seems, because of still-high inflation). Another is that Trump would do  better on immigration , another highly visible issue. A third is that most Americans ( a remarkable 81% of us ) think that Biden is too old to run for president. (Most of us think Trump is too old, too, but fewer of us.)

In a diverse country, 81% is a startling and resounding consensus.

What’s more,  76% of us have either "major" or "moderate" concerns  about Biden’s mental and physical fitness for a second term.

Now, maybe all of that is just unfairness or perception. Maybe Americans who think those things are just being stupid or falling for oppositional propaganda or are just being worrywarts.

But here’s the thing:

When it comes to elections,  perception is reality .

And, for most voters, our current  perception  is that Trump is the less bad and less old of two weak candidates that most Americans have deep misgivings about.

If, despite Biden’s age, Americans agreed he was doing a bang-up job as president or were thumping Trump in the polls, we could dismiss these concerns.

But only  37% of us approve  of the job Biden is doing.

And Biden is now  trailing ex-Pres. Trump in the swing-state polls and losing to him in the betting markets .

True — the election is nine months away.

True — a lot might change before then, including the polls.

But the way things stand today, it looks to many like Biden’s decision to accept the Democratic nomination might affect the future of America’s democracy — and, with it, the Constitution and country that Biden has sworn to protect.

Of course, a key factor in deciding whether to step aside is whether there is  anyone else  the Democrats could nominate who could be a  stronger  candidate.

Polls suggest, for example, that it would be stupid for Biden to step aside so the Democrats can nominate Vice President Kamala Harris because she  would do even worse  against Trump.

But most potential Democratic candidates deferred to Biden's desire to enter the primaries and didn't enter the race, so I don't think we really know yet whether there is a stronger candidate. By nominating someone at the convention, moreover, Democrats can choose the candidate they think has the best chance to win the general election , as opposed to the one that appeals most to Democratic voters in the primaries.

As New York Times columnist Ross Douthat  has explained , it would be better if Biden waited until mid-summer to make (or at least announce) his decision:

The next six months would provide time to see whether the polls change meaningfully — a key factor in the analysis. The waiting period would also give the broader Democratic party time to figure out whether anyone else might make a stronger candidate. And it would allow more of Trump's legal trials to play out.

If, by early August, the analysis suggests that Biden is, in fact, the strongest candidate, then the best way to serve his country would be to accept the nomination. If, on the other hand, Biden concludes that someone else might have a better chance, he could serve his country better by dropping out.

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  • x^4-5x^2+4=0
  • \sqrt{x-1}-x=-7
  • \left|3x+1\right|=4
  • \log _2(x+1)=\log _3(27)
  • 3^x=9^{x+5}
  • What is the completing square method?
  • Completing the square method is a technique for find the solutions of a quadratic equation of the form ax^2 + bx + c = 0. This method involves completing the square of the quadratic expression to the form (x + d)^2 = e, where d and e are constants.
  • What is the golden rule for solving equations?
  • The golden rule for solving equations is to keep both sides of the equation balanced so that they are always equal.
  • How do you simplify equations?
  • To simplify equations, combine like terms, remove parethesis, use the order of operations.
  • How do you solve linear equations?
  • To solve a linear equation, get the variable on one side of the equation by using inverse operations.

equation-calculator

  • High School Math Solutions – Quadratic Equations Calculator, Part 1 A quadratic equation is a second degree polynomial having the general form ax^2 + bx + c = 0, where a, b, and c... Read More

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