Algebra Examples

$2 \leq \frac{4 x + 2}{- 3} \leq 10$

Step 1

Factor $2$ out of $4 x + 2$ .

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Step 1.1

Factor $2$ out of $4 x$ .

$2 \leq \frac{2 (2 x) + 2}{- 3} \leq 10$

Step 1.2

Factor $2$ out of $2$ .

$2 \leq \frac{2 (2 x) + 2 (1)}{- 3} \leq 10$

Step 1.3

Factor $2$ out of $2 (2 x) + 2 (1)$ .

$2 \leq \frac{2 (2 x + 1)}{- 3} \leq 10$

Step 2

Move the negative in front of the fraction.

$2 \leq - \frac{2 (2 x + 1)}{3} \leq 10$

Step 3

Multiply each term in the inequality by $3$ .

$2 \cdot 3 \leq - \frac{2 (2 x + 1)}{3} \cdot 3 \leq 10 \cdot 3$

Step 4

Multiply $2$ by $3$ .

$6 \leq - \frac{2 (2 x + 1)}{3} \cdot 3 \leq 10 \cdot 3$

Step 5

Cancel the common factor of $3$ .

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Step 5.1

Move the leading negative in $- \frac{2 (2 x + 1)}{3}$ into the numerator.

$6 \leq \frac{- 2 (2 x + 1)}{3} \cdot 3 \leq 10 \cdot 3$

Step 5.2

Cancel the common factor.

$6 \leq \frac{- 2 (2 x + 1)}{3} \cdot 3 \leq 10 \cdot 3$

Step 5.3

Rewrite the expression.

$6 \leq - 2 (2 x + 1) \leq 10 \cdot 3$

Step 6

Apply the distributive property.

$6 \leq - 2 (2 x) - 2 \cdot 1 \leq 10 \cdot 3$

Step 7

Multiply.

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Step 7.1

Multiply $2$ by $- 2$ .

$6 \leq - 4 x - 2 \cdot 1 \leq 10 \cdot 3$

Step 7.2

Multiply $- 2$ by $1$ .

$6 \leq - 4 x - 2 \leq 10 \cdot 3$

Step 8

Multiply $10$ by $3$ .

$6 \leq - 4 x - 2 \leq 30$

Step 9

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

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Step 9.1

Add $2$ to each section of the inequality because it does not contain the variable we are trying to solve for.

$6 + 2 \leq - 4 x \leq 30 + 2$

Step 9.2

Add $6$ and $2$ .

$8 \leq - 4 x \leq 30 + 2$

Step 9.3

Add $30$ and $2$ .

$8 \leq - 4 x \leq 32$

Step 10

Divide each term in the inequality by $- 4$ .

$\frac{8}{- 4} \geq \frac{- 4 x}{- 4} \geq \frac{32}{- 4}$

Step 11

Divide $8$ by $- 4$ .

$- 2 \geq \frac{- 4 x}{- 4} \geq \frac{32}{- 4}$

Step 12

Cancel the common factor of $- 4$ .

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Step 12.1

Cancel the common factor.

$- 2 \geq \frac{- 4 x}{- 4} \geq \frac{32}{- 4}$

Step 12.2

Divide $x$ by $1$ .

$- 2 \geq x \geq \frac{32}{- 4}$

Step 13

Divide $32$ by $- 4$ .

$- 2 \geq x \geq - 8$

Step 14

Rewrite the interval so that the left value is less than the right value. This is the correct way to write an interval solution.

$- 8 \leq x \leq - 2$

Step 15

The result can be shown in multiple forms.

Inequality Form:

$- 8 \leq x \leq - 2$

Interval Notation:

$[- 8, - 2]$

Step 16