Algebra Examples

$(2 x - 5) (3 x + 1) {(x - 5)}^{2} \leq 0$

Step 1

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

$2 x - 5 = 0$

$3 x + 1 = 0$

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

Step 2

Set $2 x - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.1

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Step 2.2

Solve $2 x - 5 = 0$ for $x$ .

Tap for more steps...

Step 2.2.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 2.2.2

Divide each term in $2 x = 5$ by $2$ and simplify.

Tap for more steps...

Step 2.2.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 2.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 3

Set $3 x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 3.2

Solve $3 x + 1 = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 3.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

Tap for more steps...

Step 3.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 3.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 3.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 4

Set ${(x - 5)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set ${(x - 5)}^{2}$ equal to $0$ .

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

Step 4.2

Solve ${(x - 5)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

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

$x - 5 = 0$

Step 4.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 5

The final solution is all the values that make $(2 x - 5) (3 x + 1) {(x - 5)}^{2} \leq 0$ true.

$x = \frac{5}{2}, - \frac{1}{3}, 5$

Step 6

Use each root to create test intervals.

$x < - \frac{1}{3}$

$- \frac{1}{3} < x < \frac{5}{2}$

$\frac{5}{2} < x < 5$

$x > 5$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 7.1

Test a value on the interval $x < - \frac{1}{3}$ to see if it makes the inequality true.

Tap for more steps...

Step 7.1.1

Choose a value on the interval $x < - \frac{1}{3}$ and see if this value makes the original inequality true.

$x = - 3$

Step 7.1.2

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

$(2 (- 3) - 5) (3 (- 3) + 1) {((- 3) - 5)}^{2} \leq 0$

Step 7.1.3

The left side $5632$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $- \frac{1}{3} < x < \frac{5}{2}$ to see if it makes the inequality true.

Tap for more steps...

Step 7.2.1

Choose a value on the interval $- \frac{1}{3} < x < \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 7.2.2

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

$(2 (0) - 5) (3 (0) + 1) {((0) - 5)}^{2} \leq 0$

Step 7.2.3

The left side $- 125$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $\frac{5}{2} < x < 5$ to see if it makes the inequality true.

Tap for more steps...

Step 7.3.1

Choose a value on the interval $\frac{5}{2} < x < 5$ and see if this value makes the original inequality true.

$x = 4$

Step 7.3.2

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

$(2 (4) - 5) (3 (4) + 1) {((4) - 5)}^{2} \leq 0$

Step 7.3.3

The left side $39$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 7.4

Test a value on the interval $x > 5$ to see if it makes the inequality true.

Tap for more steps...

Step 7.4.1

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Step 7.4.2

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

$(2 (8) - 5) (3 (8) + 1) {((8) - 5)}^{2} \leq 0$

Step 7.4.3

The left side $2475$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 7.5

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

$x < - \frac{1}{3}$ False

$- \frac{1}{3} < x < \frac{5}{2}$ True

$\frac{5}{2} < x < 5$ False

$x > 5$ False

$x < - \frac{1}{3}$ False

$- \frac{1}{3} < x < \frac{5}{2}$ True

$\frac{5}{2} < x < 5$ False

$x > 5$ False

Step 8

The solution consists of all of the true intervals.

$- \frac{1}{3} \leq x \leq \frac{5}{2}$ or $x = 5$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- \frac{1}{3} \leq x \leq \frac{5}{2} or x = 5$

Interval Notation:

$[- \frac{1}{3}, \frac{5}{2}] \cup [5, 5]$

Step 10