Algebra Examples

$\sqrt{4 x^{2} - 12 x + 9} \leq 9$

Step 1

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

${\sqrt{4 x^{2} - 12 x + 9}}^{2} \leq 9^{2}$

Step 2

Simplify each side of the inequality.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x^{2} - 12 x + 9}$ as ${(4 x^{2} - 12 x + 9)}^{\frac{1}{2}}$ .

${({(4 x^{2} - 12 x + 9)}^{\frac{1}{2}})}^{2} \leq 9^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(4 x^{2} - 12 x + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(4 x^{2} - 12 x + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(4 x^{2} - 12 x + 9)}^{\frac{1}{2} \cdot 2} \leq 9^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(4 x^{2} - 12 x + 9)}^{\frac{1}{2} \cdot 2} \leq 9^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(4 x^{2} - 12 x + 9)}^{1} \leq 9^{2}$

Step 2.2.1.2

Simplify.

$4 x^{2} - 12 x + 9 \leq 9^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$4 x^{2} - 12 x + 9 \leq 81$

Step 3

Solve for $x$ .

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Step 3.1

Convert the inequality to an equation.

$4 x^{2} - 12 x + 9 = 81$

Step 3.2

Subtract $81$ from both sides of the equation.

$4 x^{2} - 12 x + 9 - 81 = 0$

Step 3.3

Subtract $81$ from $9$ .

$4 x^{2} - 12 x - 72 = 0$

Step 3.4

Factor the left side of the equation.

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Step 3.4.1

Factor $4$ out of $4 x^{2} - 12 x - 72$ .

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Step 3.4.1.1

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) - 12 x - 72 = 0$

Step 3.4.1.2

Factor $4$ out of $- 12 x$ .

$4 (x^{2}) + 4 (- 3 x) - 72 = 0$

Step 3.4.1.3

Factor $4$ out of $- 72$ .

$4 x^{2} + 4 (- 3 x) + 4 \cdot - 18 = 0$

Step 3.4.1.4

Factor $4$ out of $4 x^{2} + 4 (- 3 x)$ .

$4 (x^{2} - 3 x) + 4 \cdot - 18 = 0$

Step 3.4.1.5

Factor $4$ out of $4 (x^{2} - 3 x) + 4 \cdot - 18$ .

$4 (x^{2} - 3 x - 18) = 0$

Step 3.4.2

Factor.

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Step 3.4.2.1

Factor $x^{2} - 3 x - 18$ using the AC method.

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Step 3.4.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 3.4.2.1.2

Write the factored form using these integers.

$4 ((x - 6) (x + 3)) = 0$

Step 3.4.2.2

Remove unnecessary parentheses.

$4 (x - 6) (x + 3) = 0$

Step 3.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 6 = 0$

$x + 3 = 0$

Step 3.6

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.6.2

Add $6$ to both sides of the equation.

$x = 6$

Step 3.7

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.7.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.8

The final solution is all the values that make $4 (x - 6) (x + 3) = 0$ true.

$x = 6, - 3$

Step 4

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 6$

$x > 6$

Step 5

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 5.1

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

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Step 5.1.1

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

$x = - 6$

Step 5.1.2

Replace $x$ with $- 6$ in the original inequality.

$\sqrt{4 {(- 6)}^{2} - 12 \cdot - 6 + 9} \leq 9$

Step 5.1.3

The left side $15$ is greater than the right side $9$ , which means that the given statement is false.

False

Step 5.2

Test a value on the interval $- 3 < x < 6$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $- 3 < x < 6$ and see if this value makes the original inequality true.

$x = 0$

Step 5.2.2

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

$\sqrt{4 {(0)}^{2} - 12 \cdot 0 + 9} \leq 9$

Step 5.2.3

The left side $3$ is less than the right side $9$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $x > 6$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

Step 5.3.2

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

$\sqrt{4 {(8)}^{2} - 12 \cdot 8 + 9} \leq 9$

Step 5.3.3

The left side $13$ is greater than the right side $9$ , which means that the given statement is false.

False

Step 5.4

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

$x < - 3$ False

$- 3 < x < 6$ True

$x > 6$ False

$x < - 3$ False

$- 3 < x < 6$ True

$x > 6$ False

Step 6

The solution consists of all of the true intervals.

$- 3 \leq x \leq 6$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$- 3 \leq x \leq 6$

Interval Notation:

$[- 3, 6]$

Step 8