Graph x^2+4x-15<x+3

$x^{2} + 4 x - 15 < x + 3$

Step 1

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

Subtract $x$ from both sides of the inequality.

$x^{2} + 4 x - 15 - x < 3$

Subtract $x$ from $4 x$ .

$x^{2} + 3 x - 15 < 3$

Step 2

Convert the inequality to an equation.

$x^{2} + 3 x - 15 = 3$

Step 3

Subtract $3$ from both sides of the equation.

$x^{2} + 3 x - 15 - 3 = 0$

Step 4

Subtract $3$ from $- 15$ .

$x^{2} + 3 x - 18 = 0$

Step 5

Factor $x^{2} + 3 x - 18$ using the AC method.

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 $- 18$ and whose sum is $3$ .

$- 3, 6$

Write the factored form using these integers.

$(x - 3) (x + 6) = 0$

Step 6

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

$x - 3 = 0$

$x + 6 = 0$

Step 7

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 8

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 9

The final solution is all the values that make $(x - 3) (x + 6) = 0$ true.

$x = 3, - 6$

Step 10

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 3$

$x > 3$

Step 11

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

Test a value on the interval $x < - 6$ to see if it makes the inequality true.

Choose a value on the interval $x < - 6$ and see if this value makes the original inequality true.

$x = - 8$

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

${(- 8)}^{2} + 4 (- 8) - 15 < (- 8) + 3$

The left side $17$ is not less than the right side $- 5$ , which means that the given statement is false.

False

Test a value on the interval $- 6 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $- 6 < x < 3$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} + 4 (0) - 15 < (0) + 3$

The left side $- 15$ is less than the right side $3$ , which means that the given statement is always true.

True

Test a value on the interval $x > 3$ to see if it makes the inequality true.

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

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

${(6)}^{2} + 4 (6) - 15 < (6) + 3$

The left side $45$ is not less than the right side $9$ , which means that the given statement is false.

False

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

$x < - 6$ False

$- 6 < x < 3$ True

$x > 3$ False

$x < - 6$ False

$- 6 < x < 3$ True

$x > 3$ False

Step 12

The solution consists of all of the true intervals.

$- 6 < x < 3$

Step 13

Graph x^2+4x-15<x+3

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