Solve the Inequality for x (5x-6)^2<=0

${(5 x - 6)}^{2} \leq 0$

Step 1

Take the square root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{{(5 x - 6)}^{2}} \leq \sqrt{0}$

Step 2

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| 5 x - 6 | \leq \sqrt{0}$

Simplify the right side.

Simplify $\sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$| 5 x - 6 | \leq \sqrt{0^{2}}$

Pull terms out from under the radical.

$| 5 x - 6 | \leq | 0 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| 5 x - 6 | \leq 0$

Step 3

Write $| 5 x - 6 | \leq 0$ as a piecewise.

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

$5 x - 6 \geq 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$5 x \geq 6$

Divide each term in $5 x \geq 6$ by $5$ and simplify.

Divide each term in $5 x \geq 6$ by $5$ .

$\frac{5 x}{5} \geq \frac{6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{6}{5}$

Divide $x$ by $1$ .

$x \geq \frac{6}{5}$

In the piece where $5 x - 6$ is non-negative, remove the absolute value.

$5 x - 6 \leq 0$

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

$5 x - 6 < 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$5 x < 6$

Divide each term in $5 x < 6$ by $5$ and simplify.

Divide each term in $5 x < 6$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

In the piece where $5 x - 6$ is negative, remove the absolute value and multiply by $- 1$ .

$- (5 x - 6) \leq 0$

Write as a piecewise.

${\begin{cases} 5 x - 6 \leq 0 & x \geq \frac{6}{5} \\ - (5 x - 6) \leq 0 & x < \frac{6}{5} \end{cases}$

Simplify $- (5 x - 6) \leq 0$ .

Apply the distributive property.

${\begin{cases} 5 x - 6 \leq 0 & x \geq \frac{6}{5} \\ - (5 x) - - 6 \leq 0 & x < \frac{6}{5} \end{cases}$

Multiply $5$ by $- 1$ .

${\begin{cases} 5 x - 6 \leq 0 & x \geq \frac{6}{5} \\ - 5 x - - 6 \leq 0 & x < \frac{6}{5} \end{cases}$

Multiply $- 1$ by $- 6$ .

${\begin{cases} 5 x - 6 \leq 0 & x \geq \frac{6}{5} \\ - 5 x + 6 \leq 0 & x < \frac{6}{5} \end{cases}$

Step 4

Solve $5 x - 6 \leq 0$ when $x \geq \frac{6}{5}$ .

Solve $5 x - 6 \leq 0$ for $x$ .

Add $6$ to both sides of the inequality.

$5 x \leq 6$

Divide each term in $5 x \leq 6$ by $5$ and simplify.

Divide each term in $5 x \leq 6$ by $5$ .

$\frac{5 x}{5} \leq \frac{6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} \leq \frac{6}{5}$

Divide $x$ by $1$ .

$x \leq \frac{6}{5}$

Find the intersection of $x \leq \frac{6}{5}$ and $x \geq \frac{6}{5}$ .

$x = \frac{6}{5}$

Step 5

Solve $- 5 x + 6 \leq 0$ when $x < \frac{6}{5}$ .

Solve $- 5 x + 6 \leq 0$ for $x$ .

Subtract $6$ from both sides of the inequality.

$- 5 x \leq - 6$

Divide each term in $- 5 x \leq - 6$ by $- 5$ and simplify.

Divide each term in $- 5 x \leq - 6$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} \geq \frac{- 6}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} \geq \frac{- 6}{- 5}$

Divide $x$ by $1$ .

$x \geq \frac{- 6}{- 5}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \geq \frac{6}{5}$

Find the intersection of $x \geq \frac{6}{5}$ and $x < \frac{6}{5}$ .

No solution

Step 6

Find the union of the solutions.

$x = \frac{6}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{6}{5}$

Decimal Form:

$x = 1.2$

Mixed Number Form:

$x = 1 \frac{1}{5}$

Step 8

Solve the Inequality for x (5x-6)^2<=0

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