Precalculus Examples

$- 8 \leq 3 x + 6 \leq \frac{1}{2}$

Step 1

Move all terms not containing $x$ from the center section of the inequality.

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Step 1.1

Subtract $6$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$- 8 - 6 \leq 3 x \leq \frac{1}{2} - 6$

Step 1.2

Subtract $6$ from $- 8$ .

$- 14 \leq 3 x \leq \frac{1}{2} - 6$

Step 1.3

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 14 \leq 3 x \leq \frac{1}{2} - 6 \cdot \frac{2}{2}$

Step 1.4

Combine $- 6$ and $\frac{2}{2}$ .

$- 14 \leq 3 x \leq \frac{1}{2} + \frac{- 6 \cdot 2}{2}$

Step 1.5

Combine the numerators over the common denominator.

$- 14 \leq 3 x \leq \frac{1 - 6 \cdot 2}{2}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 6$ by $2$ .

$- 14 \leq 3 x \leq \frac{1 - 12}{2}$

Step 1.6.2

Subtract $12$ from $1$ .

$- 14 \leq 3 x \leq \frac{- 11}{2}$

Step 1.7

Move the negative in front of the fraction.

$- 14 \leq 3 x \leq - \frac{11}{2}$

Step 2

Divide each term in the inequality by $3$ .

$\frac{- 14}{3} \leq \frac{3 x}{3} \leq \frac{- \frac{11}{2}}{3}$

Step 3

Move the negative in front of the fraction.

$- \frac{14}{3} \leq \frac{3 x}{3} \leq \frac{- \frac{11}{2}}{3}$

Step 4

Cancel the common factor of $3$ .

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Step 4.1

Cancel the common factor.

$- \frac{14}{3} \leq \frac{3 x}{3} \leq \frac{- \frac{11}{2}}{3}$

Step 4.2

Divide $x$ by $1$ .

$- \frac{14}{3} \leq x \leq \frac{- \frac{11}{2}}{3}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$- \frac{14}{3} \leq x \leq - \frac{11}{2} \cdot \frac{1}{3}$

Step 6

Multiply $- \frac{11}{2} \cdot \frac{1}{3}$ .

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Step 6.1

Multiply $\frac{1}{3}$ by $\frac{11}{2}$ .

$- \frac{14}{3} \leq x \leq - \frac{11}{3 \cdot 2}$

Step 6.2

Multiply $3$ by $2$ .

$- \frac{14}{3} \leq x \leq - \frac{11}{6}$

Step 7

Convert the inequality to interval notation.

$[- \frac{14}{3}, - \frac{11}{6}]$

Step 8