Basic Math Examples

$20 x^{2} - 9 x \geq 20$

Step 1

Convert the inequality to an equation.

$20 x^{2} - 9 x = 20$

Step 2

Subtract $20$ from both sides of the equation.

$20 x^{2} - 9 x - 20 = 0$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 20 \cdot - 20 = - 400$ and whose sum is $b = - 9$ .

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Step 3.1.1

Factor $- 9$ out of $- 9 x$ .

$20 x^{2} - 9 x - 20 = 0$

Step 3.1.2

Rewrite $- 9$ as $16$ plus $- 25$

$20 x^{2} + (16 - 25) x - 20 = 0$

Step 3.1.3

Apply the distributive property.

$20 x^{2} + 16 x - 25 x - 20 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(20 x^{2} + 16 x) - 25 x - 20 = 0$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$4 x (5 x + 4) - 5 (5 x + 4) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 4$ .

$(5 x + 4) (4 x - 5) = 0$

Step 4

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

$5 x + 4 = 0$

$4 x - 5 = 0$

Step 5

Set $5 x + 4$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $5 x + 4$ equal to $0$ .

$5 x + 4 = 0$

Step 5.2

Solve $5 x + 4 = 0$ for $x$ .

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Step 5.2.1

Subtract $4$ from both sides of the equation.

$5 x = - 4$

Step 5.2.2

Divide each term in $5 x = - 4$ by $5$ and simplify.

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Step 5.2.2.1

Divide each term in $5 x = - 4$ by $5$ .

$\frac{5 x}{5} = \frac{- 4}{5}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 4}{5}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{5}$

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.

$x = - \frac{4}{5}$

Step 6

Set $4 x - 5$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $4 x - 5$ equal to $0$ .

$4 x - 5 = 0$

Step 6.2

Solve $4 x - 5 = 0$ for $x$ .

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Step 6.2.1

Add $5$ to both sides of the equation.

$4 x = 5$

Step 6.2.2

Divide each term in $4 x = 5$ by $4$ and simplify.

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Step 6.2.2.1

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{4}$

Step 7

The final solution is all the values that make $(5 x + 4) (4 x - 5) = 0$ true.

$x = - \frac{4}{5}, \frac{5}{4}$

Step 8

Use each root to create test intervals.

$x < - \frac{4}{5}$

$- \frac{4}{5} < x < \frac{5}{4}$

$x > \frac{5}{4}$

Step 9

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 9.1

Test a value on the interval $x < - \frac{4}{5}$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < - \frac{4}{5}$ and see if this value makes the original inequality true.

$x = - 3$

Step 9.1.2

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

$20 {(- 3)}^{2} - 9 \cdot - 3 \geq 20$

Step 9.1.3

The left side $207$ is greater than the right side $20$ , which means that the given statement is always true.

True

Step 9.2

Test a value on the interval $- \frac{4}{5} < x < \frac{5}{4}$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- \frac{4}{5} < x < \frac{5}{4}$ and see if this value makes the original inequality true.

$x = 0$

Step 9.2.2

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

$20 {(0)}^{2} - 9 \cdot 0 \geq 20$

Step 9.2.3

The left side $0$ is less than the right side $20$ , which means that the given statement is false.

False

Step 9.3

Test a value on the interval $x > \frac{5}{4}$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $x > \frac{5}{4}$ and see if this value makes the original inequality true.

$x = 4$

Step 9.3.2

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

$20 {(4)}^{2} - 9 \cdot 4 \geq 20$

Step 9.3.3

The left side $284$ is greater than the right side $20$ , which means that the given statement is always true.

True

Step 9.4

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

$x < - \frac{4}{5}$ True

$- \frac{4}{5} < x < \frac{5}{4}$ False

$x > \frac{5}{4}$ True

$x < - \frac{4}{5}$ True

$- \frac{4}{5} < x < \frac{5}{4}$ False

$x > \frac{5}{4}$ True

Step 10

The solution consists of all of the true intervals.

$x \leq - \frac{4}{5}$ or $x \geq \frac{5}{4}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{4}{5} or x \geq \frac{5}{4}$

Interval Notation:

$(- \infty, - \frac{4}{5}] \cup [\frac{5}{4}, \infty)$

Step 12