Algebra Examples

$5 \sqrt{x - 3} - 2 \leq 13$

Step 1

Move all terms not containing $5 \sqrt{x - 3}$ to the right side of the inequality.

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Step 1.1

Add $2$ to both sides of the inequality.

$5 \sqrt{x - 3} \leq 13 + 2$

Step 1.2

Add $13$ and $2$ .

$5 \sqrt{x - 3} \leq 15$

Step 2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${(5 \sqrt{x - 3})}^{2} \leq 15^{2}$

Step 3

Simplify each side of the inequality.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

${(5 {(x - 3)}^{\frac{1}{2}})}^{2} \leq 15^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(5 {(x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $5 {(x - 3)}^{\frac{1}{2}}$ .

$5^{2} {({(x - 3)}^{\frac{1}{2}})}^{2} \leq 15^{2}$

Step 3.2.1.2

Raise $5$ to the power of $2$ .

$25 {({(x - 3)}^{\frac{1}{2}})}^{2} \leq 15^{2}$

Step 3.2.1.3

Multiply the exponents in ${({(x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$25 {(x - 3)}^{\frac{1}{2} \cdot 2} \leq 15^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$25 {(x - 3)}^{\frac{1}{2} \cdot 2} \leq 15^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$25 {(x - 3)}^{1} \leq 15^{2}$

Step 3.2.1.4

Simplify.

$25 (x - 3) \leq 15^{2}$

Step 3.2.1.5

Apply the distributive property.

$25 x + 25 \cdot - 3 \leq 15^{2}$

Step 3.2.1.6

Multiply $25$ by $- 3$ .

$25 x - 75 \leq 15^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $15$ to the power of $2$ .

$25 x - 75 \leq 225$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Add $75$ to both sides of the inequality.

$25 x \leq 225 + 75$

Step 4.1.2

Add $225$ and $75$ .

$25 x \leq 300$

Step 4.2

Divide each term in $25 x \leq 300$ by $25$ and simplify.

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Step 4.2.1

Divide each term in $25 x \leq 300$ by $25$ .

$\frac{25 x}{25} \leq \frac{300}{25}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $25$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{25 x}{25} \leq \frac{300}{25}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{300}{25}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $300$ by $25$ .

$x \leq 12$

Step 5

Find the domain of $5 \sqrt{x - 3} - 15$ .

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Step 5.1

Set the radicand in $\sqrt{x - 3}$ greater than or equal to $0$ to find where the expression is defined.

$x - 3 \geq 0$

Step 5.2

Add $3$ to both sides of the inequality.

$x \geq 3$

Step 5.3

The domain is all values of $x$ that make the expression defined.

$[3, \infty)$

Step 6

Use each root to create test intervals.

$x < 3$

$3 < x < 12$

$x > 12$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.1

Test a value on the interval $x < 3$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < 3$ and see if this value makes the original inequality true.

$x = 0$

Step 7.1.2

Replace $x$ with $0$ in the original inequality.

$5 \sqrt{(0) - 3} - 2 \leq 13$

Step 7.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 7.2

Test a value on the interval $3 < x < 12$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $3 < x < 12$ and see if this value makes the original inequality true.

$x = 8$

Step 7.2.2

Replace $x$ with $8$ in the original inequality.

$5 \sqrt{(8) - 3} - 2 \leq 13$

Step 7.2.3

The left side $9.18033988$ is less than the right side $13$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $x > 12$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $x > 12$ and see if this value makes the original inequality true.

$x = 14$

Step 7.3.2

Replace $x$ with $14$ in the original inequality.

$5 \sqrt{(14) - 3} - 2 \leq 13$

Step 7.3.3

The left side $14.58312395$ is greater than the right side $13$ , which means that the given statement is false.

False

Step 7.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 3$ False

$3 < x < 12$ True

$x > 12$ False

$x < 3$ False

$3 < x < 12$ True

$x > 12$ False

Step 8

The solution consists of all of the true intervals.

$3 \leq x \leq 12$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$3 \leq x \leq 12$

Interval Notation:

$[3, 12]$

Step 10