Solve for x |4-5x|>=6

$| 4 - 5 x | \geq 6$

Step 1

Write $| 4 - 5 x | \geq 6$ as a piecewise.

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

$4 - 5 x \geq 0$

Solve the inequality.

Subtract $4$ from both sides of the inequality.

$- 5 x \geq - 4$

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

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

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

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Dividing two negative values results in a positive value.

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

In the piece where $4 - 5 x$ is non-negative, remove the absolute value.

$4 - 5 x \geq 6$

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

$4 - 5 x < 0$

Solve the inequality.

Subtract $4$ from both sides of the inequality.

$- 5 x < - 4$

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

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

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

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Dividing two negative values results in a positive value.

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

In the piece where $4 - 5 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 - 5 x) \geq 6$

Write as a piecewise.

${\begin{cases} 4 - 5 x \geq 6 & x \leq \frac{4}{5} \\ - (4 - 5 x) \geq 6 & x > \frac{4}{5} \end{cases}$

Simplify $- (4 - 5 x) \geq 6$ .

Apply the distributive property.

${\begin{cases} 4 - 5 x \geq 6 & x \leq \frac{4}{5} \\ - 1 \cdot 4 - (- 5 x) \geq 6 & x > \frac{4}{5} \end{cases}$

Multiply $- 1$ by $4$ .

${\begin{cases} 4 - 5 x \geq 6 & x \leq \frac{4}{5} \\ - 4 - (- 5 x) \geq 6 & x > \frac{4}{5} \end{cases}$

Multiply $- 5$ by $- 1$ .

${\begin{cases} 4 - 5 x \geq 6 & x \leq \frac{4}{5} \\ - 4 + 5 x \geq 6 & x > \frac{4}{5} \end{cases}$

Step 2

Solve $4 - 5 x \geq 6$ for $x$ .

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

Subtract $4$ from both sides of the inequality.

$- 5 x \geq 6 - 4$

Subtract $4$ from $6$ .

$- 5 x \geq 2$

Divide each term in $- 5 x \geq 2$ by $- 5$ and simplify.

Divide each term in $- 5 x \geq 2$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 3

Solve $- 4 + 5 x \geq 6$ for $x$ .

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

Add $4$ to both sides of the inequality.

$5 x \geq 6 + 4$

Add $6$ and $4$ .

$5 x \geq 10$

Divide each term in $5 x \geq 10$ by $5$ and simplify.

Divide each term in $5 x \geq 10$ by $5$ .

$\frac{5 x}{5} \geq \frac{10}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{10}{5}$

Divide $x$ by $1$ .

$x \geq \frac{10}{5}$

Simplify the right side.

Divide $10$ by $5$ .

$x \geq 2$

Step 4

Find the union of the solutions.

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

Step 5

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(- \infty, - \frac{2}{5}] \cup [2, \infty)$

Step 6

Solve for x |4-5x|>=6

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