Solve the Inequality for x -5x^2+3>=-8x-1

$- 5 x^{2} + 3 \geq - 8 x - 1$

Step 1

Add $8 x$ to both sides of the inequality.

$- 5 x^{2} + 3 + 8 x \geq - 1$

Step 2

Convert the inequality to an equation.

$- 5 x^{2} + 3 + 8 x = - 1$

Step 3

Add $1$ to both sides of the equation.

$- 5 x^{2} + 3 + 8 x + 1 = 0$

Step 4

Add $3$ and $1$ .

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

Step 5

Factor the left side of the equation.

Factor $- 1$ out of $- 5 x^{2} + 8 x + 4$ .

Factor $- 1$ out of $- 5 x^{2}$ .

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

Factor $- 1$ out of $8 x$ .

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

Rewrite $4$ as $- 1 (- 4)$ .

$- (5 x^{2}) - (- 8 x) - 1 \cdot - 4 = 0$

Factor $- 1$ out of $- (5 x^{2}) - (- 8 x)$ .

$- (5 x^{2} - 8 x) - 1 \cdot - 4 = 0$

Factor $- 1$ out of $- (5 x^{2} - 8 x) - 1 (- 4)$ .

$- (5 x^{2} - 8 x - 4) = 0$

Factor.

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 $a \cdot c = 5 \cdot - 4 = - 20$ and whose sum is $b = - 8$ .

Factor $- 8$ out of $- 8 x$ .

$- (5 x^{2} - 8 x - 4) = 0$

Rewrite $- 8$ as $2$ plus $- 10$

$- (5 x^{2} + (2 - 10) x - 4) = 0$

Apply the distributive property.

$- (5 x^{2} + 2 x - 10 x - 4) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- ((5 x^{2} + 2 x) - 10 x - 4) = 0$

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

$- (x (5 x + 2) - 2 (5 x + 2)) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 x + 2$ .

$- ((5 x + 2) (x - 2)) = 0$

Remove unnecessary parentheses.

$- (5 x + 2) (x - 2) = 0$

Step 6

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

$5 x + 2 = 0$

$x - 2 = 0$

Step 7

Set $5 x + 2$ equal to $0$ and solve for $x$ .

Set $5 x + 2$ equal to $0$ .

$5 x + 2 = 0$

Solve $5 x + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Divide each term in $5 x = - 2$ by $5$ and simplify.

Divide each term in $5 x = - 2$ by $5$ .

$\frac{5 x}{5} = \frac{- 2}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 2}{5}$

Divide $x$ by $1$ .

$x = \frac{- 2}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Step 8

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 9

The final solution is all the values that make $- (5 x + 2) (x - 2) = 0$ true.

$x = - \frac{2}{5}, 2$

Step 10

Use each root to create test intervals.

$x < - \frac{2}{5}$

$- \frac{2}{5} < x < 2$

$x > 2$

Step 11

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

Test a value on the interval $x < - \frac{2}{5}$ to see if it makes the inequality true.

Choose a value on the interval $x < - \frac{2}{5}$ and see if this value makes the original inequality true.

$x = - 3$

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

$- 5 {(- 3)}^{2} + 3 \geq - 8 \cdot - 3 - 1$

The left side $- 42$ is less than the right side $23$ , which means that the given statement is false.

False

Test a value on the interval $- \frac{2}{5} < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{2}{5} < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$- 5 {(0)}^{2} + 3 \geq - 8 \cdot 0 - 1$

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

True

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

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

$- 5 {(4)}^{2} + 3 \geq - 8 \cdot 4 - 1$

The left side $- 77$ is less than the right side $- 33$ , which means that the given statement is false.

False

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

$x < - \frac{2}{5}$ False

$- \frac{2}{5} < x < 2$ True

$x > 2$ False

$x < - \frac{2}{5}$ False

$- \frac{2}{5} < x < 2$ True

$x > 2$ False

Step 12

The solution consists of all of the true intervals.

$- \frac{2}{5} \leq x \leq 2$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- \frac{2}{5} \leq x \leq 2$

Interval Notation:

$[- \frac{2}{5}, 2]$

Step 14

Solve the Inequality for x -5x^2+3>=-8x-1

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