Systems of Inequalities

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Algebra › Systems of Inequalities

Questions 1 - 10

1

Solve for :

None of the other answers are correct.

Explanation

Subtract 4 from both sides. Then subtract 9x:

Next divide both sides by -6. Don't forget to switch the inequality because of the negative sign!

2

Solve for :

None of the other answers are correct.

Explanation

Subtract 4 from both sides. Then subtract 9x:

Next divide both sides by -6. Don't forget to switch the inequality because of the negative sign!

3

What is a possible valid value of ?

Explanation

This inequality can be rewritten as:

4_x_ + 14 > 30 OR 4_x_ + 14 < –30

Solve each for x:

4_x_ + 14 > 30; 4_x_ > 16; x > 4

4_x_ + 14 < –30; 4_x_ < –44; x < –11

Therefore, anything between –11 and 4 (inclusive) will not work. Hence, the answer is 7.

4

What is a possible valid value of ?

Explanation

This inequality can be rewritten as:

4_x_ + 14 > 30 OR 4_x_ + 14 < –30

Solve each for x:

4_x_ + 14 > 30; 4_x_ > 16; x > 4

4_x_ + 14 < –30; 4_x_ < –44; x < –11

Therefore, anything between –11 and 4 (inclusive) will not work. Hence, the answer is 7.

5

Find the solution set for :

$-\frac{7}{10} < -\frac{2}{5} x \leq \frac{1}{10}$

$\left [ - \frac{1}{4},\frac{7}{4} \right )$

$\left ( - \frac{1}{4},\frac{7}{4} \right ]$

$\left ( - \frac{7}{4},\frac{1}{4} \right ]$

$\left [ \frac{1}{4},\frac{7}{4} \right )$

$\left [ -\frac{7}{4},-\frac{1}{4} \right )$

Explanation

$-\frac{7}{10} < -\frac{2}{5} x \leq \frac{1}{10}$

$- \frac{5}{2} \cdot \left (-\frac{7}{10} \right )> - \frac{5}{2} \cdot \left ( -\frac{2}{5}x \right ) \geq - \frac{5}{2} \cdot \frac{1}{10}$

Note the switch in inequality symbols when the numbers are multiplied by a negative number.

$\frac{5}{2} \cdot \frac{7}{10} > x \geq - \frac{5}{2} \cdot \frac{1}{10}$

Cross-cancel:

$\frac{1}{2} \cdot \frac{7}{2} > x \geq - \frac{1}{2} \cdot \frac{1}{2}$

$\frac{7}{4} > x \geq - \frac{1}{4}$

or, in interval form, $\left [ - \frac{1}{4},\frac{7}{4} \right )$

6

Find the solution set for :

$-\frac{7}{10} < -\frac{2}{5} x \leq \frac{1}{10}$

$\left [ - \frac{1}{4},\frac{7}{4} \right )$

$\left ( - \frac{1}{4},\frac{7}{4} \right ]$

$\left ( - \frac{7}{4},\frac{1}{4} \right ]$

$\left [ \frac{1}{4},\frac{7}{4} \right )$

$\left [ -\frac{7}{4},-\frac{1}{4} \right )$

Explanation

$-\frac{7}{10} < -\frac{2}{5} x \leq \frac{1}{10}$

$- \frac{5}{2} \cdot \left (-\frac{7}{10} \right )> - \frac{5}{2} \cdot \left ( -\frac{2}{5}x \right ) \geq - \frac{5}{2} \cdot \frac{1}{10}$

Note the switch in inequality symbols when the numbers are multiplied by a negative number.

$\frac{5}{2} \cdot \frac{7}{10} > x \geq - \frac{5}{2} \cdot \frac{1}{10}$

Cross-cancel:

$\frac{1}{2} \cdot \frac{7}{2} > x \geq - \frac{1}{2} \cdot \frac{1}{2}$

$\frac{7}{4} > x \geq - \frac{1}{4}$

or, in interval form, $\left [ - \frac{1}{4},\frac{7}{4} \right )$

7

Solve for .

$5 + 3p \leq 2p + 12$

$p\leq 7$

$p\leq \frac{19}{5}$

$p\leq-7$

$p\leq17$

$p\leq\frac{7}{5}$

Explanation

First subtract 2p from both sides:

p + 5 < 12.

Then subtract 5 from both sides:

p < 7

8

Solve for .

$5 + 3p \leq 2p + 12$

$p\leq 7$

$p\leq \frac{19}{5}$

$p\leq-7$

$p\leq17$

$p\leq\frac{7}{5}$

Explanation

First subtract 2p from both sides:

p + 5 < 12.

Then subtract 5 from both sides:

p < 7

9

Solve:

$\small x+2\geq 7$

$\small x\geq 5$

$\small x\leq 9$

$\small x> 5$

$\small x< 9$

None of the other answers are correct.

Explanation

Subtract 2 from each side:

$\small x+2-2\geq 7-2$

$\small x\geq 5$

10

Solve:

$\small x+2\geq 7$

$\small x\geq 5$

$\small x\leq 9$

$\small x> 5$

$\small x< 9$

None of the other answers are correct.

Explanation

Subtract 2 from each side:

$\small x+2-2\geq 7-2$

$\small x\geq 5$

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