Algebra Examples

$\sqrt{3 x + 4} - 5 \leq 4$

Step 1

Move all terms not containing $\sqrt{3 x + 4}$ to the right side of the inequality.

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Step 1.1

Add $5$ to both sides of the inequality.

$\sqrt{3 x + 4} \leq 4 + 5$

Step 1.2

Add $4$ and $5$ .

$\sqrt{3 x + 4} \leq 9$

Step 2

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

${\sqrt{3 x + 4}}^{2} \leq 9^{2}$

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 $\sqrt{3 x + 4}$ as ${(3 x + 4)}^{\frac{1}{2}}$ .

${({(3 x + 4)}^{\frac{1}{2}})}^{2} \leq 9^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(3 x + 4)}^{\frac{1}{2} \cdot 2} \leq 9^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(3 x + 4)}^{\frac{1}{2} \cdot 2} \leq 9^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(3 x + 4)}^{1} \leq 9^{2}$

Step 3.2.1.2

Simplify.

$3 x + 4 \leq 9^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $9$ to the power of $2$ .

$3 x + 4 \leq 81$

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

Subtract $4$ from both sides of the inequality.

$3 x \leq 81 - 4$

Step 4.1.2

Subtract $4$ from $81$ .

$3 x \leq 77$

Step 4.2

Divide each term in $3 x \leq 77$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 x \leq 77$ by $3$ .

$\frac{3 x}{3} \leq \frac{77}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \leq \frac{77}{3}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{77}{3}$

Step 5

Find the domain of $\sqrt{3 x + 4} - 9$ .

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Step 5.1

Set the radicand in $\sqrt{3 x + 4}$ greater than or equal to $0$ to find where the expression is defined.

$3 x + 4 \geq 0$

Step 5.2

Solve for $x$ .

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Step 5.2.1

Subtract $4$ from both sides of the inequality.

$3 x \geq - 4$

Step 5.2.2

Divide each term in $3 x \geq - 4$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 x \geq - 4$ by $3$ .

$\frac{3 x}{3} \geq \frac{- 4}{3}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{- 4}{3}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 4}{3}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{4}{3}$

Step 5.3

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

$[- \frac{4}{3}, \infty)$

Step 6

Use each root to create test intervals.

$x < - \frac{4}{3}$

$- \frac{4}{3} < x < \frac{77}{3}$

$x > \frac{77}{3}$

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 $x < - \frac{4}{3}$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < - \frac{4}{3}$ and see if this value makes the original inequality true.

$x = - 4$

Step 7.1.2

Replace $x$ with $- 4$ in the original inequality.

$\sqrt{3 (- 4) + 4} - 5 \leq 4$

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 $- \frac{4}{3} < x < \frac{77}{3}$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- \frac{4}{3} < x < \frac{77}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 7.2.2

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

$\sqrt{3 (0) + 4} - 5 \leq 4$

Step 7.2.3

The left side $- 3$ is less than the right side $4$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $x > \frac{77}{3}$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $x > \frac{77}{3}$ and see if this value makes the original inequality true.

$x = 28$

Step 7.3.2

Replace $x$ with $28$ in the original inequality.

$\sqrt{3 (28) + 4} - 5 \leq 4$

Step 7.3.3

The left side $4.38083151$ is greater than the right side $4$ , which means that the given statement is false.

False

Step 7.4

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

$x < - \frac{4}{3}$ False

$- \frac{4}{3} < x < \frac{77}{3}$ True

$x > \frac{77}{3}$ False

$x < - \frac{4}{3}$ False

$- \frac{4}{3} < x < \frac{77}{3}$ True

$x > \frac{77}{3}$ False

Step 8

The solution consists of all of the true intervals.

$- \frac{4}{3} \leq x \leq \frac{77}{3}$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- \frac{4}{3} \leq x \leq \frac{77}{3}$

Interval Notation:

$[- \frac{4}{3}, \frac{77}{3}]$

Step 10