Algebra Examples

$- 4 x^{2} - 13 > 11 x - 7$

Step 1

Subtract $11 x$ from both sides of the inequality.

$- 4 x^{2} - 13 - 11 x > - 7$

Step 2

Convert the inequality to an equation.

$- 4 x^{2} - 13 - 11 x = - 7$

Step 3

Add $7$ to both sides of the equation.

$- 4 x^{2} - 13 - 11 x + 7 = 0$

Step 4

Add $- 13$ and $7$ .

$- 4 x^{2} - 11 x - 6 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $- 1$ out of $- 4 x^{2} - 11 x - 6$ .

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Step 5.1.1

Factor $- 1$ out of $- 4 x^{2}$ .

$- (4 x^{2}) - 11 x - 6 = 0$

Step 5.1.2

Factor $- 1$ out of $- 11 x$ .

$- (4 x^{2}) - (11 x) - 6 = 0$

Step 5.1.3

Rewrite $- 6$ as $- 1 (6)$ .

$- (4 x^{2}) - (11 x) - 1 \cdot 6 = 0$

Step 5.1.4

Factor $- 1$ out of $- (4 x^{2}) - (11 x)$ .

$- (4 x^{2} + 11 x) - 1 \cdot 6 = 0$

Step 5.1.5

Factor $- 1$ out of $- (4 x^{2} + 11 x) - 1 (6)$ .

$- (4 x^{2} + 11 x + 6) = 0$

Step 5.2

Factor.

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Step 5.2.1

Factor by grouping.

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Step 5.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 6 = 24$ and whose sum is $b = 11$ .

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Step 5.2.1.1.1

Factor $11$ out of $11 x$ .

$- (4 x^{2} + 11 (x) + 6) = 0$

Step 5.2.1.1.2

Rewrite $11$ as $3$ plus $8$

$- (4 x^{2} + (3 + 8) x + 6) = 0$

Step 5.2.1.1.3

Apply the distributive property.

$- (4 x^{2} + 3 x + 8 x + 6) = 0$

Step 5.2.1.2

Factor out the greatest common factor from each group.

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Step 5.2.1.2.1

Group the first two terms and the last two terms.

$- ((4 x^{2} + 3 x) + 8 x + 6) = 0$

Step 5.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (x (4 x + 3) + 2 (4 x + 3)) = 0$

Step 5.2.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 3$ .

$- ((4 x + 3) (x + 2)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$- (4 x + 3) (x + 2) = 0$

Step 6

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

$4 x + 3 = 0$

$x + 2 = 0$

Step 7

Set $4 x + 3$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Step 7.2

Solve $4 x + 3 = 0$ for $x$ .

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Step 7.2.1

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Step 7.2.2

Divide each term in $4 x = - 3$ by $4$ and simplify.

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Step 7.2.2.1

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $4$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

Step 8

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 8.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 9

The final solution is all the values that make $- (4 x + 3) (x + 2) = 0$ true.

$x = - \frac{3}{4}, - 2$

Step 10

Use each root to create test intervals.

$x < - 2$

$- 2 < x < - \frac{3}{4}$

$x > - \frac{3}{4}$

Step 11

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 11.1

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

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Step 11.1.1

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Step 11.1.2

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

$- 4 {(- 4)}^{2} - 13 > 11 (- 4) - 7$

Step 11.1.3

The left side $- 77$ is not greater than the right side $- 51$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $- 2 < x < - \frac{3}{4}$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $- 2 < x < - \frac{3}{4}$ and see if this value makes the original inequality true.

$x = - 1$

Step 11.2.2

Replace $x$ with $- 1$ in the original inequality.

$- 4 {(- 1)}^{2} - 13 > 11 (- 1) - 7$

Step 11.2.3

The left side $- 17$ is greater than the right side $- 18$ , which means that the given statement is always true.

True

Step 11.3

Test a value on the interval $x > - \frac{3}{4}$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $x > - \frac{3}{4}$ and see if this value makes the original inequality true.

$x = 0$

Step 11.3.2

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

$- 4 {(0)}^{2} - 13 > 11 (0) - 7$

Step 11.3.3

The left side $- 13$ is not greater than the right side $- 7$ , which means that the given statement is false.

False

Step 11.4

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

$x < - 2$ False

$- 2 < x < - \frac{3}{4}$ True

$x > - \frac{3}{4}$ False

$x < - 2$ False

$- 2 < x < - \frac{3}{4}$ True

$x > - \frac{3}{4}$ False

Step 12

The solution consists of all of the true intervals.

$- 2 < x < - \frac{3}{4}$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < - \frac{3}{4}$

Interval Notation:

$(- 2, - \frac{3}{4})$

Step 14