Algebra Examples

$4 x^{2} - 4 x - 4 > - 1$

Step 1

Convert the inequality to an equation.

$4 x^{2} - 4 x - 4 = - 1$

Step 2

Add $1$ to both sides of the equation.

$4 x^{2} - 4 x - 4 + 1 = 0$

Step 3

Add $- 4$ and $1$ .

$4 x^{2} - 4 x - 3 = 0$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 4 \cdot - 3 = - 12$ and whose sum is $b = - 4$ .

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Step 4.1.1

Factor $- 4$ out of $- 4 x$ .

$4 x^{2} - 4 x - 3 = 0$

Step 4.1.2

Rewrite $- 4$ as $2$ plus $- 6$

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

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

Step 4.2.2

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

$2 x (2 x + 1) - 3 (2 x + 1) = 0$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(2 x + 1) (2 x - 3) = 0$

Step 5

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

$2 x + 1 = 0$

$2 x - 3 = 0$

Step 6

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 6.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 6.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 7

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 7.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 7.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 7.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 7.2.2.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 8

The final solution is all the values that make $(2 x + 1) (2 x - 3) = 0$ true.

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

Step 9

Use each root to create test intervals.

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

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

$x > \frac{3}{2}$

Step 10

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 10.1

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

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Step 10.1.1

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

$x = - 3$

Step 10.1.2

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

$4 {(- 3)}^{2} - 4 \cdot - 3 - 4 > - 1$

Step 10.1.3

The left side $44$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

Step 10.2

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

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Step 10.2.1

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

$x = 0$

Step 10.2.2

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

$4 {(0)}^{2} - 4 \cdot 0 - 4 > - 1$

Step 10.2.3

The left side $- 4$ is not greater than the right side $- 1$ , which means that the given statement is false.

False

Step 10.3

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

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Step 10.3.1

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

$x = 4$

Step 10.3.2

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

$4 {(4)}^{2} - 4 \cdot 4 - 4 > - 1$

Step 10.3.3

The left side $44$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

Step 10.4

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

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

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

$x > \frac{3}{2}$ True

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

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

$x > \frac{3}{2}$ True

Step 11

The solution consists of all of the true intervals.

$x < - \frac{1}{2}$ or $x > \frac{3}{2}$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{1}{2} or x > \frac{3}{2}$

Interval Notation:

$(- \infty, - \frac{1}{2}) \cup (\frac{3}{2}, \infty)$

Step 13