Algebra Examples

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

Step 1

Take the specified 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.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Pull terms out from under the radical.

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\sqrt{0}$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

Pull terms out from under the radical.

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

Step 2.2.1.3

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.

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Step 3.1

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

$5 x - 6 \geq 0$

Step 3.2

Solve the inequality.

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Step 3.2.1

Add $6$ to both sides of the inequality.

$5 x \geq 6$

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.3

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

$5 x - 6 \leq 0$

Step 3.4

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

$5 x - 6 < 0$

Step 3.5

Solve the inequality.

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Step 3.5.1

Add $6$ to both sides of the inequality.

$5 x < 6$

Step 3.5.2

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

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Step 3.5.2.1

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

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

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $5$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

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

Step 3.5.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.6

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

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

Step 3.7

Write as a piecewise.

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

Step 3.8

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

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Multiply $5$ by $- 1$ .

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

Step 3.8.3

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

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Step 4.1

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

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Step 4.1.1

Add $6$ to both sides of the inequality.

$5 x \leq 6$

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

Simplify the left side.

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Step 4.1.2.2.1

Cancel the common factor of $5$ .

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Step 4.1.2.2.1.1

Cancel the common factor.

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

Step 4.1.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.2

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

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Step 5.1

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

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Step 5.1.1

Subtract $6$ from both sides of the inequality.

$- 5 x \leq - 6$

Step 5.1.2

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

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Step 5.1.2.1

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.

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

Step 5.1.2.2

Simplify the left side.

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Step 5.1.2.2.1

Cancel the common factor of $- 5$ .

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Step 5.1.2.2.1.1

Cancel the common factor.

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

Step 5.1.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.1.2.3

Simplify the right side.

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Step 5.1.2.3.1

Dividing two negative values results in a positive value.

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

Step 5.2

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