Solve for x |6-4x|<=1

$| 6 - 4 x | \leq 1$

Step 1

Write $| 6 - 4 x | \leq 1$ as a piecewise.

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

$6 - 4 x \geq 0$

Solve the inequality.

Subtract $6$ from both sides of the inequality.

$- 4 x \geq - 6$

Divide each term in $- 4 x \geq - 6$ by $- 4$ and simplify.

Divide each term in $- 4 x \geq - 6$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \leq \frac{- 6}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} \leq \frac{- 6}{- 4}$

Divide $x$ by $1$ .

$x \leq \frac{- 6}{- 4}$

Simplify the right side.

Cancel the common factor of $- 6$ and $- 4$ .

Factor $- 2$ out of $- 6$ .

$x \leq \frac{- 2 (3)}{- 4}$

Cancel the common factors.

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

$x \leq \frac{- 2 \cdot 3}{- 2 \cdot 2}$

Cancel the common factor.

$x \leq \frac{- 2 \cdot 3}{- 2 \cdot 2}$

Rewrite the expression.

$x \leq \frac{3}{2}$

In the piece where $6 - 4 x$ is non-negative, remove the absolute value.

$6 - 4 x \leq 1$

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

$6 - 4 x < 0$

Solve the inequality.

Subtract $6$ from both sides of the inequality.

$- 4 x < - 6$

Divide each term in $- 4 x < - 6$ by $- 4$ and simplify.

Divide each term in $- 4 x < - 6$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} > \frac{- 6}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} > \frac{- 6}{- 4}$

Divide $x$ by $1$ .

$x > \frac{- 6}{- 4}$

Simplify the right side.

Cancel the common factor of $- 6$ and $- 4$ .

Factor $- 2$ out of $- 6$ .

$x > \frac{- 2 (3)}{- 4}$

Cancel the common factors.

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

$x > \frac{- 2 \cdot 3}{- 2 \cdot 2}$

Cancel the common factor.

$x > \frac{- 2 \cdot 3}{- 2 \cdot 2}$

Rewrite the expression.

$x > \frac{3}{2}$

In the piece where $6 - 4 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (6 - 4 x) \leq 1$

Write as a piecewise.

${\begin{cases} 6 - 4 x \leq 1 & x \leq \frac{3}{2} \\ - (6 - 4 x) \leq 1 & x > \frac{3}{2} \end{cases}$

Simplify $- (6 - 4 x) \leq 1$ .

Apply the distributive property.

${\begin{cases} 6 - 4 x \leq 1 & x \leq \frac{3}{2} \\ - 1 \cdot 6 - (- 4 x) \leq 1 & x > \frac{3}{2} \end{cases}$

Multiply $- 1$ by $6$ .

${\begin{cases} 6 - 4 x \leq 1 & x \leq \frac{3}{2} \\ - 6 - (- 4 x) \leq 1 & x > \frac{3}{2} \end{cases}$

Multiply $- 4$ by $- 1$ .

${\begin{cases} 6 - 4 x \leq 1 & x \leq \frac{3}{2} \\ - 6 + 4 x \leq 1 & x > \frac{3}{2} \end{cases}$

Step 2

Solve $6 - 4 x \leq 1$ when $x \leq \frac{3}{2}$ .

Solve $6 - 4 x \leq 1$ for $x$ .

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

Subtract $6$ from both sides of the inequality.

$- 4 x \leq 1 - 6$

Subtract $6$ from $1$ .

$- 4 x \leq - 5$

Divide each term in $- 4 x \leq - 5$ by $- 4$ and simplify.

Divide each term in $- 4 x \leq - 5$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \geq \frac{- 5}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} \geq \frac{- 5}{- 4}$

Divide $x$ by $1$ .

$x \geq \frac{- 5}{- 4}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \geq \frac{5}{4}$

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

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

Step 3

Solve $- 6 + 4 x \leq 1$ when $x > \frac{3}{2}$ .

Solve $- 6 + 4 x \leq 1$ for $x$ .

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

Add $6$ to both sides of the inequality.

$4 x \leq 1 + 6$

Add $1$ and $6$ .

$4 x \leq 7$

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 $x > \frac{3}{2}$ .

$\frac{3}{2} < x \leq \frac{7}{4}$

Step 4

Find the union of the solutions.

$\frac{5}{4} \leq x \leq \frac{7}{4}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$\frac{5}{4} \leq x \leq \frac{7}{4}$

Interval Notation:

$[\frac{5}{4}, \frac{7}{4}]$

Step 6

Solve for x |6-4x|<=1

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