Solve the Inequality for x 6x^2-30x<-7x+4

$6 x^{2} - 30 x < - 7 x + 4$

Step 1

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

Add $7 x$ to both sides of the inequality.

$6 x^{2} - 30 x + 7 x < 4$

Add $- 30 x$ and $7 x$ .

$6 x^{2} - 23 x < 4$

Step 2

Convert the inequality to an equation.

$6 x^{2} - 23 x = 4$

Step 3

Subtract $4$ from both sides of the equation.

$6 x^{2} - 23 x - 4 = 0$

Step 4

Factor by grouping.

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 $b = - 23$ .

Factor $- 23$ out of $- 23 x$ .

$6 x^{2} - 23 x - 4 = 0$

Rewrite $- 23$ as $1$ plus $- 24$

$6 x^{2} + (1 - 24) x - 4 = 0$

Apply the distributive property.

$6 x^{2} + 1 x - 24 x - 4 = 0$

Multiply $x$ by $1$ .

$6 x^{2} + x - 24 x - 4 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 x^{2} + x) - 24 x - 4 = 0$

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

$x (6 x + 1) - 4 (6 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $6 x + 1$ .

$(6 x + 1) (x - 4) = 0$

Step 5

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

$6 x + 1 = 0$

$x - 4 = 0$

Step 6

Set $6 x + 1$ equal to $0$ and solve for $x$ .

Set $6 x + 1$ equal to $0$ .

$6 x + 1 = 0$

Solve $6 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$6 x = - 1$

Divide each term in $6 x = - 1$ by $6$ and simplify.

Divide each term in $6 x = - 1$ by $6$ .

$\frac{6 x}{6} = \frac{- 1}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 1}{6}$

Divide $x$ by $1$ .

$x = \frac{- 1}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{6}$

Step 7

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 8

The final solution is all the values that make $(6 x + 1) (x - 4) = 0$ true.

$x = - \frac{1}{6}, 4$

Step 9

Use each root to create test intervals.

$x < - \frac{1}{6}$

$- \frac{1}{6} < x < 4$

$x > 4$

Step 10

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 $x < - \frac{1}{6}$ to see if it makes the inequality true.

Choose a value on the interval $x < - \frac{1}{6}$ and see if this value makes the original inequality true.

$x = - 3$

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

$6 {(- 3)}^{2} - 30 \cdot - 3 < - 7 \cdot - 3 + 4$

The left side $144$ is not less than the right side $25$ , which means that the given statement is false.

False

Test a value on the interval $- \frac{1}{6} < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{1}{6} < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

$6 {(0)}^{2} - 30 \cdot 0 < - 7 \cdot 0 + 4$

The left side $0$ is less than the right side $4$ , which means that the given statement is always true.

True

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

$6 {(6)}^{2} - 30 \cdot 6 < - 7 \cdot 6 + 4$

The left side $36$ is not less than the right side $- 38$ , which means that the given statement is false.

False

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

$x < - \frac{1}{6}$ False

$- \frac{1}{6} < x < 4$ True

$x > 4$ False

$x < - \frac{1}{6}$ False

$- \frac{1}{6} < x < 4$ True

$x > 4$ False

Step 11

The solution consists of all of the true intervals.

$- \frac{1}{6} < x < 4$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- \frac{1}{6} < x < 4$

Interval Notation:

$(- \frac{1}{6}, 4)$

Step 13

Solve the Inequality for x 6x^2-30x<-7x+4

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