Solve for x 5(x^2-1)>-24x

$5 (x^{2} - 1) > - 24 x$

Step 1

Simplify $5 (x^{2} - 1)$ .

Rewrite.

$0 + 0 + 5 (x^{2} - 1) > - 24 x$

Simplify by adding zeros.

$5 (x^{2} - 1) > - 24 x$

Apply the distributive property.

$5 x^{2} + 5 \cdot - 1 > - 24 x$

Multiply $5$ by $- 1$ .

$5 x^{2} - 5 > - 24 x$

Step 2

Add $24 x$ to both sides of the inequality.

$5 x^{2} - 5 + 24 x > 0$

Step 3

Convert the inequality to an equation.

$5 x^{2} - 5 + 24 x = 0$

Step 4

Factor by grouping.

Reorder terms.

$5 x^{2} + 24 x - 5 = 0$

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 $a \cdot c = 5 \cdot - 5 = - 25$ and whose sum is $b = 24$ .

Factor $24$ out of $24 x$ .

$5 x^{2} + 24 (x) - 5 = 0$

Rewrite $24$ as $- 1$ plus $25$

$5 x^{2} + (- 1 + 25) x - 5 = 0$

Apply the distributive property.

$5 x^{2} - 1 x + 25 x - 5 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(5 x^{2} - 1 x) + 25 x - 5 = 0$

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

$x (5 x - 1) + 5 (5 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 x - 1$ .

$(5 x - 1) (x + 5) = 0$

Step 5

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

$5 x - 1 = 0$

$x + 5 = 0$

Step 6

Set $5 x - 1$ equal to $0$ and solve for $x$ .

Set $5 x - 1$ equal to $0$ .

$5 x - 1 = 0$

Solve $5 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$5 x = 1$

Divide each term in $5 x = 1$ by $5$ and simplify.

Divide each term in $5 x = 1$ by $5$ .

$\frac{5 x}{5} = \frac{1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{1}{5}$

Divide $x$ by $1$ .

$x = \frac{1}{5}$

Step 7

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 8

The final solution is all the values that make $(5 x - 1) (x + 5) = 0$ true.

$x = \frac{1}{5}, - 5$

Step 9

Use each root to create test intervals.

$x < - 5$

$- 5 < x < \frac{1}{5}$

$x > \frac{1}{5}$

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 $x < - 5$ to see if it makes the inequality true.

Choose a value on the interval $x < - 5$ and see if this value makes the original inequality true.

$x = - 8$

Replace $x$ with $- 8$ in the original inequality.

$5 ({(- 8)}^{2} - 1) > - 24 \cdot - 8$

The left side $315$ is greater than the right side $192$ , which means that the given statement is always true.

True

Test a value on the interval $- 5 < x < \frac{1}{5}$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < \frac{1}{5}$ and see if this value makes the original inequality true.

$x = 0$

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

$5 ({(0)}^{2} - 1) > - 24 \cdot 0$

The left side $- 5$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > \frac{1}{5}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{1}{5}$ and see if this value makes the original inequality true.

$x = 3$

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

$5 ({(3)}^{2} - 1) > - 24 \cdot 3$

The left side $40$ is greater than the right side $- 72$ , which means that the given statement is always true.

True

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

$x < - 5$ True

$- 5 < x < \frac{1}{5}$ False

$x > \frac{1}{5}$ True

$x < - 5$ True

$- 5 < x < \frac{1}{5}$ False

$x > \frac{1}{5}$ True

Step 11

The solution consists of all of the true intervals.

$x < - 5$ or $x > \frac{1}{5}$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x < - 5 or x > \frac{1}{5}$

Interval Notation:

$(- \infty, - 5) \cup (\frac{1}{5}, \infty)$

Step 13

Solve for x 5(x^2-1)>-24x

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