Algebra Examples

$- 8 \leq \frac{- 3 x + 1}{4} \leq 11$

Step 1

Factor $- 1$ out of $- 3 x$ .

$- 8 \leq \frac{- (3 x) + 1}{4} \leq 11$

Step 2

Rewrite $1$ as $- 1 (- 1)$ .

$- 8 \leq \frac{- (3 x) - 1 \cdot - 1}{4} \leq 11$

Step 3

Factor $- 1$ out of $- (3 x) - 1 (- 1)$ .

$- 8 \leq \frac{- (3 x - 1)}{4} \leq 11$

Step 4

Simplify the expression.

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Step 4.1

Rewrite $- (3 x - 1)$ as $- 1 (3 x - 1)$ .

$- 8 \leq \frac{- 1 (3 x - 1)}{4} \leq 11$

Step 4.2

Move the negative in front of the fraction.

$- 8 \leq - \frac{3 x - 1}{4} \leq 11$

Step 5

Multiply each term in the inequality by $4$ .

$- 8 \cdot 4 \leq - \frac{3 x - 1}{4} \cdot 4 \leq 11 \cdot 4$

Step 6

Multiply $- 8$ by $4$ .

$- 32 \leq - \frac{3 x - 1}{4} \cdot 4 \leq 11 \cdot 4$

Step 7

Cancel the common factor of $4$ .

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Step 7.1

Move the leading negative in $- \frac{3 x - 1}{4}$ into the numerator.

$- 32 \leq \frac{- (3 x - 1)}{4} \cdot 4 \leq 11 \cdot 4$

Step 7.2

Cancel the common factor.

$- 32 \leq \frac{- (3 x - 1)}{4} \cdot 4 \leq 11 \cdot 4$

Step 7.3

Rewrite the expression.

$- 32 \leq - (3 x - 1) \leq 11 \cdot 4$

Step 8

Apply the distributive property.

$- 32 \leq - (3 x) + 1 \leq 11 \cdot 4$

Step 9

Multiply.

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Step 9.1

Multiply $3$ by $- 1$ .

$- 32 \leq - 3 x + 1 \leq 11 \cdot 4$

Step 9.2

Multiply $- 1$ by $- 1$ .

$- 32 \leq - 3 x + 1 \leq 11 \cdot 4$

Step 10

Multiply $11$ by $4$ .

$- 32 \leq - 3 x + 1 \leq 44$

Step 11

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

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Step 11.1

Subtract $1$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$- 32 - 1 \leq - 3 x \leq 44 - 1$

Step 11.2

Subtract $1$ from $- 32$ .

$- 33 \leq - 3 x \leq 44 - 1$

Step 11.3

Subtract $1$ from $44$ .

$- 33 \leq - 3 x \leq 43$

Step 12

Divide each term in the inequality by $- 3$ .

$\frac{- 33}{- 3} \geq \frac{- 3 x}{- 3} \geq \frac{43}{- 3}$

Step 13

Divide $- 33$ by $- 3$ .

$11 \geq \frac{- 3 x}{- 3} \geq \frac{43}{- 3}$

Step 14

Cancel the common factor of $- 3$ .

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Step 14.1

Cancel the common factor.

$11 \geq \frac{- 3 x}{- 3} \geq \frac{43}{- 3}$

Step 14.2

Divide $x$ by $1$ .

$11 \geq x \geq \frac{43}{- 3}$

Step 15

Move the negative in front of the fraction.

$11 \geq x \geq - \frac{43}{3}$

Step 16

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

$- \frac{43}{3} \leq x \leq 11$

Step 17

The result can be shown in multiple forms.

Inequality Form:

$- \frac{43}{3} \leq x \leq 11$

Interval Notation:

$[- \frac{43}{3}, 11]$

Step 18