Solve the Inequality for x 6x^2-6<=5x

$6 x^{2} - 6 \leq 5 x$

Step 1

Subtract $5 x$ from both sides of the inequality.

$6 x^{2} - 6 - 5 x \leq 0$

Step 2

Convert the inequality to an equation.

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

Step 3

Factor by grouping.

Reorder terms.

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

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 $a \cdot c = 6 \cdot - 6 = - 36$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

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

Rewrite $- 5$ as $4$ plus $- 9$

$6 x^{2} + (4 - 9) x - 6 = 0$

Apply the distributive property.

$6 x^{2} + 4 x - 9 x - 6 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 x^{2} + 4 x) - 9 x - 6 = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x + 2$ .

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

Step 4

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

$3 x + 2 = 0$

$2 x - 3 = 0$

Step 5

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

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

$3 x + 2 = 0$

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

Subtract $2$ from both sides of the equation.

$3 x = - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 6

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

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

$2 x - 3 = 0$

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

Add $3$ to both sides of the equation.

$2 x = 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 7

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

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

Step 8

Use each root to create test intervals.

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

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

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

Step 9

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

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

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

$x = - 3$

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

$6 {(- 3)}^{2} - 6 \leq 5 (- 3)$

The left side $48$ is greater than the right side $- 15$ , which means that the given statement is false.

False

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

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

$x = 0$

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

$6 {(0)}^{2} - 6 \leq 5 (0)$

The left side $- 6$ is less than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 4$

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

$6 {(4)}^{2} - 6 \leq 5 (4)$

The left side $90$ is greater than the right side $20$ , which means that the given statement is false.

False

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

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

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

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

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

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

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

Step 10

The solution consists of all of the true intervals.

$- \frac{2}{3} \leq x \leq \frac{3}{2}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- \frac{2}{3} \leq x \leq \frac{3}{2}$

Interval Notation:

$[- \frac{2}{3}, \frac{3}{2}]$

Step 12

Solve the Inequality for x 6x^2-6<=5x

Related Questions

Support LCMGCF.com by disabling your adblocker.