Pre-Algebra Examples

$| 10 - 4 x | < 13$

Step 1

Write $| 10 - 4 x | < 13$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$10 - 4 x \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $10$ from both sides of the inequality.

$- 4 x \geq - 10$

Step 1.2.2

Divide each term in $- 4 x \geq - 10$ by $- 4$ and simplify.

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Step 1.2.2.1

Divide each term in $- 4 x \geq - 10$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \leq \frac{- 10}{- 4}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} \leq \frac{- 10}{- 4}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 10}{- 4}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Cancel the common factor of $- 10$ and $- 4$ .

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Step 1.2.2.3.1.1

Factor $- 2$ out of $- 10$ .

$x \leq \frac{- 2 (5)}{- 4}$

Step 1.2.2.3.1.2

Cancel the common factors.

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Step 1.2.2.3.1.2.1

Factor $- 2$ out of $- 4$ .

$x \leq \frac{- 2 \cdot 5}{- 2 \cdot 2}$

Step 1.2.2.3.1.2.2

Cancel the common factor.

$x \leq \frac{- 2 \cdot 5}{- 2 \cdot 2}$

Step 1.2.2.3.1.2.3

Rewrite the expression.

$x \leq \frac{5}{2}$

Step 1.3

In the piece where $10 - 4 x$ is non-negative, remove the absolute value.

$10 - 4 x < 13$

Step 1.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$10 - 4 x < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $10$ from both sides of the inequality.

$- 4 x < - 10$

Step 1.5.2

Divide each term in $- 4 x < - 10$ by $- 4$ and simplify.

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Step 1.5.2.1

Divide each term in $- 4 x < - 10$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} > \frac{- 10}{- 4}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} > \frac{- 10}{- 4}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 10}{- 4}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Cancel the common factor of $- 10$ and $- 4$ .

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Step 1.5.2.3.1.1

Factor $- 2$ out of $- 10$ .

$x > \frac{- 2 (5)}{- 4}$

Step 1.5.2.3.1.2

Cancel the common factors.

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Step 1.5.2.3.1.2.1

Factor $- 2$ out of $- 4$ .

$x > \frac{- 2 \cdot 5}{- 2 \cdot 2}$

Step 1.5.2.3.1.2.2

Cancel the common factor.

$x > \frac{- 2 \cdot 5}{- 2 \cdot 2}$

Step 1.5.2.3.1.2.3

Rewrite the expression.

$x > \frac{5}{2}$

Step 1.6

In the piece where $10 - 4 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (10 - 4 x) < 13$

Step 1.7

Write as a piecewise.

${\begin{cases} 10 - 4 x < 13 & x \leq \frac{5}{2} \\ - (10 - 4 x) < 13 & x > \frac{5}{2} \end{cases}$

Step 1.8

Simplify $- (10 - 4 x) < 13$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} 10 - 4 x < 13 & x \leq \frac{5}{2} \\ - 1 \cdot 10 - (- 4 x) < 13 & x > \frac{5}{2} \end{cases}$

Step 1.8.2

Multiply $- 1$ by $10$ .

${\begin{cases} 10 - 4 x < 13 & x \leq \frac{5}{2} \\ - 10 - (- 4 x) < 13 & x > \frac{5}{2} \end{cases}$

Step 1.8.3

Multiply $- 4$ by $- 1$ .

${\begin{cases} 10 - 4 x < 13 & x \leq \frac{5}{2} \\ - 10 + 4 x < 13 & x > \frac{5}{2} \end{cases}$

Step 2

Solve $10 - 4 x < 13$ when $x \leq \frac{5}{2}$ .

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Step 2.1

Solve $10 - 4 x < 13$ for $x$ .

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Step 2.1.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 2.1.1.1

Subtract $10$ from both sides of the inequality.

$- 4 x < 13 - 10$

Step 2.1.1.2

Subtract $10$ from $13$ .

$- 4 x < 3$

Step 2.1.2

Divide each term in $- 4 x < 3$ by $- 4$ and simplify.

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Step 2.1.2.1

Divide each term in $- 4 x < 3$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} > \frac{3}{- 4}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} > \frac{3}{- 4}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{3}{- 4}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Move the negative in front of the fraction.

$x > - \frac{3}{4}$

Step 2.2

Find the intersection of $x > - \frac{3}{4}$ and $x \leq \frac{5}{2}$ .

$- \frac{3}{4} < x \leq \frac{5}{2}$

Step 3

Solve $- 10 + 4 x < 13$ when $x > \frac{5}{2}$ .

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Step 3.1

Solve $- 10 + 4 x < 13$ for $x$ .

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Step 3.1.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 3.1.1.1

Add $10$ to both sides of the inequality.

$4 x < 13 + 10$

Step 3.1.1.2

Add $13$ and $10$ .

$4 x < 23$

Step 3.1.2

Divide each term in $4 x < 23$ by $4$ and simplify.

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Step 3.1.2.1

Divide each term in $4 x < 23$ by $4$ .

$\frac{4 x}{4} < \frac{23}{4}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $4$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{23}{4}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{23}{4}$

Step 3.2

Find the intersection of $x < \frac{23}{4}$ and $x > \frac{5}{2}$ .

$\frac{5}{2} < x < \frac{23}{4}$

Step 4

Find the union of the solutions.

$- \frac{3}{4} < x < \frac{23}{4}$

Step 5