Precalculus Examples

$25 x^{2} + 16 < 40 x$

Step 1

Subtract $40 x$ from both sides of the inequality.

$25 x^{2} + 16 - 40 x < 0$

Step 2

Convert the inequality to an equation.

$25 x^{2} + 16 - 40 x = 0$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rearrange terms.

$25 x^{2} - 40 x + 16 = 0$

Step 3.2

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

${(5 x)}^{2} - 40 x + 16 = 0$

Step 3.3

Rewrite $16$ as $4^{2}$ .

${(5 x)}^{2} - 40 x + 4^{2} = 0$

Step 3.4

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$40 x = 2 \cdot (5 x) \cdot 4$

Step 3.5

Rewrite the polynomial.

${(5 x)}^{2} - 2 \cdot (5 x) \cdot 4 + 4^{2} = 0$

Step 3.6

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 5 x$ and $b = 4$ .

${(5 x - 4)}^{2} = 0$

Step 4

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

$5 x - 4 = 0$

Step 5

Solve for $x$ .

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Step 5.1

Add $4$ to both sides of the equation.

$5 x = 4$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x$ by $1$ .

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

Step 6

Use each root to create test intervals.

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

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

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 $x < \frac{4}{5}$ to see if it makes the inequality true.

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Step 7.1.1

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

$x = 0$

Step 7.1.2

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

$25 {(0)}^{2} + 16 < 40 (0)$

Step 7.1.3

The left side $16$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.2

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

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Step 7.2.1

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

$x = 3$

Step 7.2.2

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

$25 {(3)}^{2} + 16 < 40 (3)$

Step 7.2.3

The left side $241$ is not less than the right side $120$ , which means that the given statement is false.

False

Step 7.3

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

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

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

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

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

Step 8

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution