Algebra Examples

$x^{2} - 4 x - 5 \leq 0$

Step 1

Convert the inequality to an equation.

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

Step 2

Factor $x^{2} - 4 x - 5$ using the AC method.

Tap for more steps...

Step 2.1

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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 2.2

Write the factored form using these integers.

$(x - 5) (x + 1) = 0$

Step 3

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

$x - 5 = 0$

$x + 1 = 0$

Step 4

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

Tap for more steps...

Step 4.1

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

$x - 5 = 0$

Step 4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 5

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

Tap for more steps...

Step 5.1

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

$x + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6

The final solution is all the values that make $(x - 5) (x + 1) = 0$ true.

$x = 5, - 1$

Step 7

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 5$

$x > 5$

Step 8

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 8.1

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

Tap for more steps...

Step 8.1.1

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

$x = - 4$

Step 8.1.2

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

${(- 4)}^{2} - 4 \cdot - 4 - 5 \leq 0$

Step 8.1.3

The left side $27$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 8.2

Test a value on the interval $- 1 < x < 5$ to see if it makes the inequality true.

Tap for more steps...

Step 8.2.1

Choose a value on the interval $- 1 < x < 5$ and see if this value makes the original inequality true.

$x = 0$

Step 8.2.2

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

${(0)}^{2} - 4 \cdot 0 - 5 \leq 0$

Step 8.2.3

The left side $- 5$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 8.3

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

Tap for more steps...

Step 8.3.1

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

$x = 8$

Step 8.3.2

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

${(8)}^{2} - 4 \cdot 8 - 5 \leq 0$

Step 8.3.3

The left side $27$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 8.4

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

$x < - 1$ False

$- 1 < x < 5$ True

$x > 5$ False

$x < - 1$ False

$- 1 < x < 5$ True

$x > 5$ False

Step 9

The solution consists of all of the true intervals.

$- 1 \leq x \leq 5$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 1 \leq x \leq 5$

Interval Notation:

$[- 1, 5]$

Step 11