Solve the Inequality for x |4-4x|<=3

$| 4 - 4 x | \leq 3$

Step 1

Write $| 4 - 4 x | \leq 3$ as a piecewise.

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

$4 - 4 x \geq 0$

Solve the inequality.

Subtract $4$ from both sides of the inequality.

$- 4 x \geq - 4$

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

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

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

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 4$ by $- 4$ .

$x \leq 1$

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

$4 - 4 x \leq 3$

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

$4 - 4 x < 0$

Solve the inequality.

Subtract $4$ from both sides of the inequality.

$- 4 x < - 4$

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 $- 4$ .

$x > 1$

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

$- (4 - 4 x) \leq 3$

Write as a piecewise.

${\begin{cases} 4 - 4 x \leq 3 & x \leq 1 \\ - (4 - 4 x) \leq 3 & x > 1 \end{cases}$

Simplify $- (4 - 4 x) \leq 3$ .

Apply the distributive property.

${\begin{cases} 4 - 4 x \leq 3 & x \leq 1 \\ - 1 \cdot 4 - (- 4 x) \leq 3 & x > 1 \end{cases}$

Multiply $- 1$ by $4$ .

${\begin{cases} 4 - 4 x \leq 3 & x \leq 1 \\ - 4 - (- 4 x) \leq 3 & x > 1 \end{cases}$

Multiply $- 4$ by $- 1$ .

${\begin{cases} 4 - 4 x \leq 3 & x \leq 1 \\ - 4 + 4 x \leq 3 & x > 1 \end{cases}$

Step 2

Solve $4 - 4 x \leq 3$ when $x \leq 1$ .

Solve $4 - 4 x \leq 3$ for $x$ .

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

Subtract $4$ from both sides of the inequality.

$- 4 x \leq 3 - 4$

Subtract $4$ from $3$ .

$- 4 x \leq - 1$

Divide each term in $- 4 x \leq - 1$ by $- 4$ and simplify.

Divide each term in $- 4 x \leq - 1$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \geq \frac{- 1}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} \geq \frac{- 1}{- 4}$

Divide $x$ by $1$ .

$x \geq \frac{- 1}{- 4}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \geq \frac{1}{4}$

Find the intersection of $x \geq \frac{1}{4}$ and $x \leq 1$ .

$\frac{1}{4} \leq x \leq 1$

Step 3

Solve $- 4 + 4 x \leq 3$ when $x > 1$ .

Solve $- 4 + 4 x \leq 3$ for $x$ .

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

Add $4$ to both sides of the inequality.

$4 x \leq 3 + 4$

Add $3$ and $4$ .

$4 x \leq 7$

Divide each term in $4 x \leq 7$ by $4$ and simplify.

Divide each term in $4 x \leq 7$ by $4$ .

$\frac{4 x}{4} \leq \frac{7}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \leq \frac{7}{4}$

Divide $x$ by $1$ .

$x \leq \frac{7}{4}$

Find the intersection of $x \leq \frac{7}{4}$ and $x > 1$ .

$1 < x \leq \frac{7}{4}$

Step 4

Find the union of the solutions.

$\frac{1}{4} \leq x \leq \frac{7}{4}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$\frac{1}{4} \leq x \leq \frac{7}{4}$

Interval Notation:

$[\frac{1}{4}, \frac{7}{4}]$

Step 6

Solve the Inequality for x |4-4x|<=3

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