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# Checking Solutions to Inequalities

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What if you were given an inequality like $-3x^3 < -81$ and told that one of its solutions was $x > 3$ ? How could you determine if that solution were correct? After completing this Concept, you'll be able to check the solutions to inequations like this one.

### Try This

For more practice solving inequalities, check out http://www.aaastudy.com/equ725x7.htm .

### Guidance

To check the solution to an inequality, we replace the variable in the inequality with the value of the solution. A solution to an inequality produces a true statement when substituted into the inequality.

#### Example A

Check that the given number is a solution to the inequality: $a = 10; \ 20a \le 250$

Solution

Replace the variable in the inequality with the given value.

$20(10) & \le 250\\200 & \le 250$

This statement is true . This means that $a = 10$ is a solution to the inequality $20a \le 250$ .

Note that $a = 10$ is not the only solution to this inequality. If we divide both sides of the inequality by 20, we can write it as $a \le 12.5$ . This means that any number less than or equal to 12.5 is also a solution to the inequality.

#### Example B

Check that the given number is a solution to the inequality: $b = -0.5; \ \frac{3 - b}{b} > -4$

Solution:

$\frac{3 - (-0.5)}{(-0.5)} & > -4\\\frac{3 + 0.5}{-0.5} & > -4\\- \frac{3.5}{0.5} & > -4\\-7 & > -4$

This statement is false . This means that $b = - 0.5$ is not a solution to the inequality $\frac{3 - b}{b} > -4$ .

#### Example C

To organize a picnic Peter needs at least two times as many hamburgers as hot dogs. He has 24 hot dogs. What is the possible number of hamburgers Peter has?

Solution:

Define

Let $x =$ number of hamburgers

Translate

Peter needs at least two times as many hamburgers as hot dogs. He has 24 hot dogs.

This means that twice the number of hot dogs is less than or equal to the number of hamburgers.

$2 \times 24 \le x, \ \text{or} \ 48 \le x$

Answer

Peter needs at least 48 hamburgers.

Check

48 hamburgers is twice the number of hot dogs. So more than 48 hamburgers is more than twice the number of hot dogs. The answer checks out .

Watch this video for help with the Examples above.

### Vocabulary

• A solution to an inequality produces a true statement when substituted into the inequality.

### Guided Practice

Check that the given number is a solution to the inequality: $x = \frac{3}{4}; \ 4x + 5 \le 8$

Solution:

$4 \left ( \frac{3}{4} \right ) + 5 & \ge 8\\3 + 5 & \ge 8\\8 & \ge 8$

This statement is true . It is true because this inequality includes an equals sign; since 8 is equal to itself, it is also “greater than or equal to” itself. This means that $x = \frac{3}{4}$ is a solution to the inequality $4x + 5 \le 8$ .

### Practice

For 1-4, check whether the given number is a solution to the corresponding inequality.

1. $x = 12; \ 2(x + 6) \le 8x$
2. $z = -9; \ 1.4z + 5.2 > 0.4z$
3. $y = 40; \ - \frac{5}{2}y + \frac{1}{2} < -18$
4. $t = 0.4; \ 80 \ge 10(3t + 2)$
5. On your new job you can be paid in one of two ways. You can either be paid $1000 per month plus 6% commission of total sales or be paid$1200 per month plus 5% commission on sales over $2000. For what amount of sales is the first option better than the second option? Assume there are always sales over$2000.

For 6-14, suppose a phone company offers a choice of three text-messaging plans. Plan A gives you unlimited text messages for $10 a month; Plan B gives you 60 text messages for$5 a month and then charges you $0.05 for each additional message; and Plan C has no monthly fee but charges you$0.10 per message.

1. If $m$ is the number of messages you send per month, write an expression for the monthly cost of each of the three plans.
2. For what values of $m$ is Plan A cheaper than Plan B?
3. For what values of $m$ is Plan A cheaper than Plan C?
4. For what values of $m$ is Plan B cheaper than Plan C?
5. For what values of $m$ is Plan A the cheapest of all? (Hint: for what values is A both cheaper than B and cheaper than C?)
6. For what values of $m$ is Plan B the cheapest of all? (Careful—for what values is B cheaper than A?)
7. For what values of $m$ is Plan C the cheapest of all?
8. If you send 30 messages per month, which plan is cheapest?
9. What is the cost of each of the three plans if you send 30 messages per month?

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