Pre-Algebra Examples

$- 9 z (z - 4) > 4 z - 3$

Step 1

Simplify $- 9 z (z - 4)$ .

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Step 1.1

Rewrite.

$0 + 0 - 9 z (z - 4) > 4 z - 3$

Step 1.2

Simplify by adding zeros.

$- 9 z (z - 4) > 4 z - 3$

Step 1.3

Apply the distributive property.

$- 9 z \cdot z - 9 z \cdot - 4 > 4 z - 3$

Step 1.4

Multiply $z$ by $z$ by adding the exponents.

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Step 1.4.1

Move $z$ .

$- 9 (z \cdot z) - 9 z \cdot - 4 > 4 z - 3$

Step 1.4.2

Multiply $z$ by $z$ .

$- 9 z^{2} - 9 z \cdot - 4 > 4 z - 3$

Step 1.5

Multiply $- 4$ by $- 9$ .

$- 9 z^{2} + 36 z > 4 z - 3$

Step 2

Move all terms containing $z$ to the left side of the inequality.

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Step 2.1

Subtract $4 z$ from both sides of the inequality.

$- 9 z^{2} + 36 z - 4 z > - 3$

Step 2.2

Subtract $4 z$ from $36 z$ .

$- 9 z^{2} + 32 z > - 3$

Step 3

Add $3$ to both sides of the inequality.

$- 9 z^{2} + 32 z + 3 > 0$

Step 4

Convert the inequality to an equation.

$- 9 z^{2} + 32 z + 3 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = - 9$ , $b = 32$ , and $c = 3$ into the quadratic formula and solve for $z$ .

$\frac{- 32 \pm \sqrt{32^{2} - 4 \cdot (- 9 \cdot 3)}}{2 \cdot - 9}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $32$ to the power of $2$ .

$z = \frac{- 32 \pm \sqrt{1024 - 4 \cdot - 9 \cdot 3}}{2 \cdot - 9}$

Step 7.1.2

Multiply $- 4 \cdot - 9 \cdot 3$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 9$ .

$z = \frac{- 32 \pm \sqrt{1024 + 36 \cdot 3}}{2 \cdot - 9}$

Step 7.1.2.2

Multiply $36$ by $3$ .

$z = \frac{- 32 \pm \sqrt{1024 + 108}}{2 \cdot - 9}$

Step 7.1.3

Add $1024$ and $108$ .

$z = \frac{- 32 \pm \sqrt{1132}}{2 \cdot - 9}$

Step 7.1.4

Rewrite $1132$ as $2^{2} \cdot 283$ .

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Step 7.1.4.1

Factor $4$ out of $1132$ .

$z = \frac{- 32 \pm \sqrt{4 (283)}}{2 \cdot - 9}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$z = \frac{- 32 \pm \sqrt{2^{2} \cdot 283}}{2 \cdot - 9}$

Step 7.1.5

Pull terms out from under the radical.

$z = \frac{- 32 \pm 2 \sqrt{283}}{2 \cdot - 9}$

Step 7.2

Multiply $2$ by $- 9$ .

$z = \frac{- 32 \pm 2 \sqrt{283}}{- 18}$

Step 7.3

Simplify $\frac{- 32 \pm 2 \sqrt{283}}{- 18}$ .

$z = \frac{16 \pm \sqrt{283}}{9}$

Step 8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $32$ to the power of $2$ .

$z = \frac{- 32 \pm \sqrt{1024 - 4 \cdot - 9 \cdot 3}}{2 \cdot - 9}$

Step 8.1.2

Multiply $- 4 \cdot - 9 \cdot 3$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 9$ .

$z = \frac{- 32 \pm \sqrt{1024 + 36 \cdot 3}}{2 \cdot - 9}$

Step 8.1.2.2

Multiply $36$ by $3$ .

$z = \frac{- 32 \pm \sqrt{1024 + 108}}{2 \cdot - 9}$

Step 8.1.3

Add $1024$ and $108$ .

$z = \frac{- 32 \pm \sqrt{1132}}{2 \cdot - 9}$

Step 8.1.4

Rewrite $1132$ as $2^{2} \cdot 283$ .

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Step 8.1.4.1

Factor $4$ out of $1132$ .

$z = \frac{- 32 \pm \sqrt{4 (283)}}{2 \cdot - 9}$

Step 8.1.4.2

Rewrite $4$ as $2^{2}$ .

$z = \frac{- 32 \pm \sqrt{2^{2} \cdot 283}}{2 \cdot - 9}$

Step 8.1.5

Pull terms out from under the radical.

$z = \frac{- 32 \pm 2 \sqrt{283}}{2 \cdot - 9}$

Step 8.2

Multiply $2$ by $- 9$ .

$z = \frac{- 32 \pm 2 \sqrt{283}}{- 18}$

Step 8.3

Simplify $\frac{- 32 \pm 2 \sqrt{283}}{- 18}$ .

$z = \frac{16 \pm \sqrt{283}}{9}$

Step 8.4

Change the $\pm$ to $+$ .

$z = \frac{16 + \sqrt{283}}{9}$

Step 9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $32$ to the power of $2$ .

$z = \frac{- 32 \pm \sqrt{1024 - 4 \cdot - 9 \cdot 3}}{2 \cdot - 9}$

Step 9.1.2

Multiply $- 4 \cdot - 9 \cdot 3$ .

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Step 9.1.2.1

Multiply $- 4$ by $- 9$ .

$z = \frac{- 32 \pm \sqrt{1024 + 36 \cdot 3}}{2 \cdot - 9}$

Step 9.1.2.2

Multiply $36$ by $3$ .

$z = \frac{- 32 \pm \sqrt{1024 + 108}}{2 \cdot - 9}$

Step 9.1.3

Add $1024$ and $108$ .

$z = \frac{- 32 \pm \sqrt{1132}}{2 \cdot - 9}$

Step 9.1.4

Rewrite $1132$ as $2^{2} \cdot 283$ .

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Step 9.1.4.1

Factor $4$ out of $1132$ .

$z = \frac{- 32 \pm \sqrt{4 (283)}}{2 \cdot - 9}$

Step 9.1.4.2

Rewrite $4$ as $2^{2}$ .

$z = \frac{- 32 \pm \sqrt{2^{2} \cdot 283}}{2 \cdot - 9}$

Step 9.1.5

Pull terms out from under the radical.

$z = \frac{- 32 \pm 2 \sqrt{283}}{2 \cdot - 9}$

Step 9.2

Multiply $2$ by $- 9$ .

$z = \frac{- 32 \pm 2 \sqrt{283}}{- 18}$

Step 9.3

Simplify $\frac{- 32 \pm 2 \sqrt{283}}{- 18}$ .

$z = \frac{16 \pm \sqrt{283}}{9}$

Step 9.4

Change the $\pm$ to $-$ .

$z = \frac{16 - \sqrt{283}}{9}$

Step 10

Consolidate the solutions.

$z = \frac{16 + \sqrt{283}}{9}, \frac{16 - \sqrt{283}}{9}$

Step 11

Use each root to create test intervals.

$z < \frac{16 - \sqrt{283}}{9}$

$\frac{16 - \sqrt{283}}{9} < z < \frac{16 + \sqrt{283}}{9}$

$z > \frac{16 + \sqrt{283}}{9}$

Step 12

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 12.1

Test a value on the interval $z < \frac{16 - \sqrt{283}}{9}$ to see if it makes the inequality true.

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Step 12.1.1

Choose a value on the interval $z < \frac{16 - \sqrt{283}}{9}$ and see if this value makes the original inequality true.

$z = - 3$

Step 12.1.2

Replace $z$ with $- 3$ in the original inequality.

$- 9 \cdot - 3 ((- 3) - 4) > 4 (- 3) - 3$

Step 12.1.3

The left side $- 189$ is not greater than the right side $- 15$ , which means that the given statement is false.

False

Step 12.2

Test a value on the interval $\frac{16 - \sqrt{283}}{9} < z < \frac{16 + \sqrt{283}}{9}$ to see if it makes the inequality true.

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Step 12.2.1

Choose a value on the interval $\frac{16 - \sqrt{283}}{9} < z < \frac{16 + \sqrt{283}}{9}$ and see if this value makes the original inequality true.

$z = 0$

Step 12.2.2

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

$- 9 \cdot 0 ((0) - 4) > 4 (0) - 3$

Step 12.2.3

The left side $0$ is greater than the right side $- 3$ , which means that the given statement is always true.

True

Step 12.3

Test a value on the interval $z > \frac{16 + \sqrt{283}}{9}$ to see if it makes the inequality true.

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Step 12.3.1

Choose a value on the interval $z > \frac{16 + \sqrt{283}}{9}$ and see if this value makes the original inequality true.

$z = 6$

Step 12.3.2

Replace $z$ with $6$ in the original inequality.

$- 9 \cdot 6 ((6) - 4) > 4 (6) - 3$

Step 12.3.3

The left side $- 108$ is not greater than the right side $21$ , which means that the given statement is false.

False

Step 12.4

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

$z < \frac{16 - \sqrt{283}}{9}$ False

$\frac{16 - \sqrt{283}}{9} < z < \frac{16 + \sqrt{283}}{9}$ True

$z > \frac{16 + \sqrt{283}}{9}$ False

$z < \frac{16 - \sqrt{283}}{9}$ False

$\frac{16 - \sqrt{283}}{9} < z < \frac{16 + \sqrt{283}}{9}$ True

$z > \frac{16 + \sqrt{283}}{9}$ False

Step 13

The solution consists of all of the true intervals.

$\frac{16 - \sqrt{283}}{9} < z < \frac{16 + \sqrt{283}}{9}$

Step 14