Algebra Examples

$6 x^{2} - 11 x - 1 < - 6 x + 3$

Step 1

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

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Step 1.1

Add $6 x$ to both sides of the inequality.

$6 x^{2} - 11 x - 1 + 6 x < 3$

Step 1.2

Add $- 11 x$ and $6 x$ .

$6 x^{2} - 5 x - 1 < 3$

Step 2

Convert the inequality to an equation.

$6 x^{2} - 5 x - 1 = 3$

Step 3

Subtract $3$ from both sides of the equation.

$6 x^{2} - 5 x - 1 - 3 = 0$

Step 4

Subtract $3$ from $- 1$ .

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

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

Subtract $1$ from 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 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.

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

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

Add $4$ to 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 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.

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

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

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

Step 11

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

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Step 11.1

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

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Step 11.1.1

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

$x = - 3$

Step 11.1.2

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

$6 {(- 3)}^{2} - 11 \cdot - 3 - 1 < - 6 \cdot - 3 + 3$

Step 11.1.3

The left side $86$ is not less than the right side $21$ , which means that the given statement is false.

False

Step 11.2

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

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Step 11.2.1

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

$x = 0$

Step 11.2.2

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

$6 {(0)}^{2} - 11 \cdot 0 - 1 < - 6 \cdot 0 + 3$

Step 11.2.3

The left side $- 1$ is less than the right side $3$ , which means that the given statement is always true.

True

Step 11.3

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

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Step 11.3.1

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

$x = 4$

Step 11.3.2

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

$6 {(4)}^{2} - 11 \cdot 4 - 1 < - 6 \cdot 4 + 3$

Step 11.3.3

The left side $51$ is not less than the right side $- 21$ , which means that the given statement is false.

False

Step 11.4

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

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

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

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

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

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

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

Step 12

The solution consists of all of the true intervals.

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

Step 13

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Step 14