Solve the Inequality for x |4x-2|+3<9

$| 4 x - 2 | + 3 < 9$

Step 1

Write $| 4 x - 2 | + 3 < 9$ as a piecewise.

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

$4 x - 2 \geq 0$

Solve the inequality.

Add $2$ to both sides of the inequality.

$4 x \geq 2$

Divide each term in $4 x \geq 2$ by $4$ and simplify.

Divide each term in $4 x \geq 2$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$x \geq \frac{2 (1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x \geq \frac{2 \cdot 1}{2 \cdot 2}$

Cancel the common factor.

$x \geq \frac{2 \cdot 1}{2 \cdot 2}$

Rewrite the expression.

$x \geq \frac{1}{2}$

In the piece where $4 x - 2$ is non-negative, remove the absolute value.

$4 x - 2 + 3 < 9$

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

$4 x - 2 < 0$

Solve the inequality.

Add $2$ to both sides of the inequality.

$4 x < 2$

Divide each term in $4 x < 2$ by $4$ and simplify.

Divide each term in $4 x < 2$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$x < \frac{2 (1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x < \frac{2 \cdot 1}{2 \cdot 2}$

Cancel the common factor.

$x < \frac{2 \cdot 1}{2 \cdot 2}$

Rewrite the expression.

$x < \frac{1}{2}$

In the piece where $4 x - 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 x - 2) + 3 < 9$

Write as a piecewise.

${\begin{cases} 4 x - 2 + 3 < 9 & x \geq \frac{1}{2} \\ - (4 x - 2) + 3 < 9 & x < \frac{1}{2} \end{cases}$

Add $- 2$ and $3$ .

${\begin{cases} 4 x + 1 < 9 & x \geq \frac{1}{2} \\ - (4 x - 2) + 3 < 9 & x < \frac{1}{2} \end{cases}$

Simplify $- (4 x - 2) + 3 < 9$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 4 x + 1 < 9 & x \geq \frac{1}{2} \\ - (4 x) - - 2 + 3 < 9 & x < \frac{1}{2} \end{cases}$

Multiply $4$ by $- 1$ .

${\begin{cases} 4 x + 1 < 9 & x \geq \frac{1}{2} \\ - 4 x - - 2 + 3 < 9 & x < \frac{1}{2} \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} 4 x + 1 < 9 & x \geq \frac{1}{2} \\ - 4 x + 2 + 3 < 9 & x < \frac{1}{2} \end{cases}$

Add $2$ and $3$ .

${\begin{cases} 4 x + 1 < 9 & x \geq \frac{1}{2} \\ - 4 x + 5 < 9 & x < \frac{1}{2} \end{cases}$

Step 2

Solve $4 x + 1 < 9$ when $x \geq \frac{1}{2}$ .

Solve $4 x + 1 < 9$ for $x$ .

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

Subtract $1$ from both sides of the inequality.

$4 x < 9 - 1$

Subtract $1$ from $9$ .

$4 x < 8$

Divide each term in $4 x < 8$ by $4$ and simplify.

Divide each term in $4 x < 8$ by $4$ .

$\frac{4 x}{4} < \frac{8}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} < \frac{8}{4}$

Divide $x$ by $1$ .

$x < \frac{8}{4}$

Simplify the right side.

Divide $8$ by $4$ .

$x < 2$

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

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

Step 3

Solve $- 4 x + 5 < 9$ when $x < \frac{1}{2}$ .

Solve $- 4 x + 5 < 9$ for $x$ .

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

Subtract $5$ from both sides of the inequality.

$- 4 x < 9 - 5$

Subtract $5$ from $9$ .

$- 4 x < 4$

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

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

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

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 $x < \frac{1}{2}$ .

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

Step 4

Find the union of the solutions.

$- 1 < x < 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 1 < x < 2$

Interval Notation:

$(- 1, 2)$

Step 6

Solve the Inequality for x |4x-2|+3<9

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