Algebra Examples

$x^{2} + 3 x < 28$

Step 1

Convert the inequality to an equation.

$x^{2} + 3 x = 28$

Step 2

Subtract $28$ from both sides of the equation.

$x^{2} + 3 x - 28 = 0$

Step 3

Factor $x^{2} + 3 x - 28$ using the AC method.

Tap for more steps...

Step 3.1

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 $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 3.2

Write the factored form using these integers.

$(x - 4) (x + 7) = 0$

Step 4

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

$x - 4 = 0$

$x + 7 = 0$

Step 5

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

Tap for more steps...

Step 5.1

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

$x - 4 = 0$

Step 5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 6

Set $x + 7$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 6.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 7

The final solution is all the values that make $(x - 4) (x + 7) = 0$ true.

$x = 4, - 7$

Step 8

Use each root to create test intervals.

$x < - 7$

$- 7 < x < 4$

$x > 4$

Step 9

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

Tap for more steps...

Step 9.1

Test a value on the interval $x < - 7$ to see if it makes the inequality true.

Tap for more steps...

Step 9.1.1

Choose a value on the interval $x < - 7$ and see if this value makes the original inequality true.

$x = - 10$

Step 9.1.2

Replace $x$ with $- 10$ in the original inequality.

${(- 10)}^{2} + 3 (- 10) < 28$

Step 9.1.3

The left side $70$ is not less than the right side $28$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $- 7 < x < 4$ to see if it makes the inequality true.

Tap for more steps...

Step 9.2.1

Choose a value on the interval $- 7 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 9.2.2

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

${(0)}^{2} + 3 (0) < 28$

Step 9.2.3

The left side $0$ is less than the right side $28$ , which means that the given statement is always true.

True

Step 9.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Tap for more steps...

Step 9.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 9.3.2

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

${(6)}^{2} + 3 (6) < 28$

Step 9.3.3

The left side $54$ is not less than the right side $28$ , which means that the given statement is false.

False

Step 9.4

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

$x < - 7$ False

$- 7 < x < 4$ True

$x > 4$ False

$x < - 7$ False

$- 7 < x < 4$ True

$x > 4$ False

Step 10

The solution consists of all of the true intervals.

$- 7 < x < 4$

Step 11

Convert the inequality to interval notation.

$(- 7, 4)$

Step 12