Solve the Absolute Value Inequality for x |10+4x|<14

$| 10 + 4 x | < 14$

Write $| 10 + 4 x | < 14$ as a piecewise.

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

$10 + 4 x \geq 0$

Solve the inequality.

Subtract $10$ from both sides of the inequality.

$4 x \geq - 10$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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

Factor $2$ out of $- 10$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x \geq \frac{- 5}{2}$

Move the negative in front of the fraction.

$x \geq - \frac{5}{2}$

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

$10 + 4 x < 14$

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

$10 + 4 x < 0$

Solve the inequality.

Subtract $10$ from both sides of the inequality.

$4 x < - 10$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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

Factor $2$ out of $- 10$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x < \frac{- 5}{2}$

Move the negative in front of the fraction.

$x < - \frac{5}{2}$

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

$- (10 + 4 x) < 14$

Write as a piecewise.

${\begin{cases} 10 + 4 x < 14 & x \geq - \frac{5}{2} \\ - (10 + 4 x) < 14 & x < - \frac{5}{2} \end{cases}$

Simplify $- (10 + 4 x) < 14$ .

Apply the distributive property.

${\begin{cases} 10 + 4 x < 14 & x \geq - \frac{5}{2} \\ - 1 \cdot 10 - (4 x) < 14 & x < - \frac{5}{2} \end{cases}$

Multiply $- 1$ by $10$ .

${\begin{cases} 10 + 4 x < 14 & x \geq - \frac{5}{2} \\ - 10 - (4 x) < 14 & x < - \frac{5}{2} \end{cases}$

Multiply $4$ by $- 1$ .

${\begin{cases} 10 + 4 x < 14 & x \geq - \frac{5}{2} \\ - 10 - 4 x < 14 & x < - \frac{5}{2} \end{cases}$

Solve $10 + 4 x < 14$ when $x \geq - \frac{5}{2}$ .

Solve $10 + 4 x < 14$ for $x$ .

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

Subtract $10$ from both sides of the inequality.

$4 x < 14 - 10$

Subtract $10$ from $14$ .

$4 x < 4$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $4$ by $4$ .

$x < 1$

Find the intersection of $x < 1$ and $x \geq - \frac{5}{2}$ .

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

Solve $- 10 - 4 x < 14$ when $x < - \frac{5}{2}$ .

Solve $- 10 - 4 x < 14$ for $x$ .

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

Add $10$ to both sides of the inequality.

$- 4 x < 14 + 10$

Add $14$ and $10$ .

$- 4 x < 24$

Divide each term in $- 4 x < 24$ by $- 4$ and simplify.

Divide each term in $- 4 x < 24$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} > \frac{24}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} > \frac{24}{- 4}$

Divide $x$ by $1$ .

$x > \frac{24}{- 4}$

Simplify the right side.

Divide $24$ by $- 4$ .

$x > - 6$

Find the intersection of $x > - 6$ and $x < - \frac{5}{2}$ .

$- 6 < x < - \frac{5}{2}$

Find the union of the solutions.

$- 6 < x < 1$

The result can be shown in multiple forms.

Inequality Form:

$- 6 < x < 1$

Interval Notation:

$(- 6, 1)$

Solve the Absolute Value Inequality for x |10+4x|<14

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