Algebra Examples

$6 x^{2} + 4 x + 4 < - x + 8$

Step 1

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

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Step 1.1

Add $x$ to both sides of the inequality.

$6 x^{2} + 4 x + 4 + x < 8$

Step 1.2

Add $4 x$ and $x$ .

$6 x^{2} + 5 x + 4 < 8$

Step 2

Convert the inequality to an equation.

$6 x^{2} + 5 x + 4 = 8$

Step 3

Subtract $8$ from both sides of the equation.

$6 x^{2} + 5 x + 4 - 8 = 0$

Step 4

Subtract $8$ from $4$ .

$6 x^{2} + 5 x - 4 = 0$

Step 5

Factor by grouping.

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Step 5.1

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 $a \cdot c = 6 \cdot - 4 = - 24$ and whose sum is $b = 5$ .

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Step 5.1.1

Factor $5$ out of $5 x$ .

$6 x^{2} + 5 (x) - 4 = 0$

Step 5.1.2

Rewrite $5$ as $- 3$ plus $8$

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

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

Step 5.2.2

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

$3 x (2 x - 1) + 4 (2 x - 1) = 0$

Step 5.3

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

$(2 x - 1) (3 x + 4) = 0$

Step 6

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

$2 x - 1 = 0$

$3 x + 4 = 0$

Step 7

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

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Step 7.1

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

$2 x - 1 = 0$

Step 7.2

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

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Step 7.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 7.2.2

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

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Step 7.2.2.1

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

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

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.

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

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 8

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

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Step 8.1

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

$3 x + 4 = 0$

Step 8.2

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

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Step 8.2.1

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Step 8.2.2

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

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Step 8.2.2.1

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

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

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $3$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 8.2.2.3

Simplify the right side.

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Step 8.2.2.3.1

Move the negative in front of the fraction.

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

Step 9

The final solution is all the values that make $(2 x - 1) (3 x + 4) = 0$ true.

$x = \frac{1}{2}, - \frac{4}{3}$

Step 10

Use each root to create test intervals.

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

$- \frac{4}{3} < x < \frac{1}{2}$

$x > \frac{1}{2}$

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 $x < - \frac{4}{3}$ to see if it makes the inequality true.

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Step 11.1.1

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

$x = - 4$

Step 11.1.2

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

$6 {(- 4)}^{2} + 4 (- 4) + 4 < - (- 4) + 8$

Step 11.1.3

The left side $84$ is not less than the right side $12$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $- \frac{4}{3} < x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $- \frac{4}{3} < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 11.2.2

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

$6 {(0)}^{2} + 4 (0) + 4 < - (0) + 8$

Step 11.2.3

The left side $4$ is less than the right side $8$ , which means that the given statement is always true.

True

Step 11.3

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

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Step 11.3.1

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

$x = 3$

Step 11.3.2

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

$6 {(3)}^{2} + 4 (3) + 4 < - (3) + 8$

Step 11.3.3

The left side $70$ is not less than the right side $5$ , which means that the given statement is false.

False

Step 11.4

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

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

$- \frac{4}{3} < x < \frac{1}{2}$ True

$x > \frac{1}{2}$ False

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

$- \frac{4}{3} < x < \frac{1}{2}$ True

$x > \frac{1}{2}$ False

Step 12

The solution consists of all of the true intervals.

$- \frac{4}{3} < x < \frac{1}{2}$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- \frac{4}{3} < x < \frac{1}{2}$

Interval Notation:

$(- \frac{4}{3}, \frac{1}{2})$

Step 14