Solve the Inequality for x 4x^2+8x-21>=0

$4 x^{2} + 8 x - 21 \geq 0$

Step 1

Convert the inequality to an equation.

$4 x^{2} + 8 x - 21 = 0$

Step 2

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 $a \cdot c = 4 \cdot - 21 = - 84$ and whose sum is $b = 8$ .

Factor $8$ out of $8 x$ .

$4 x^{2} + 8 (x) - 21 = 0$

Rewrite $8$ as $- 6$ plus $14$

$4 x^{2} + (- 6 + 14) x - 21 = 0$

Apply the distributive property.

$4 x^{2} - 6 x + 14 x - 21 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} - 6 x) + 14 x - 21 = 0$

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

$2 x (2 x - 3) + 7 (2 x - 3) = 0$

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

$(2 x - 3) (2 x + 7) = 0$

Step 3

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

$2 x - 3 = 0$

$2 x + 7 = 0$

Step 4

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 5

Set $2 x + 7$ equal to $0$ and solve for $x$ .

Set $2 x + 7$ equal to $0$ .

$2 x + 7 = 0$

Solve $2 x + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Divide each term in $2 x = - 7$ by $2$ and simplify.

Divide each term in $2 x = - 7$ by $2$ .

$\frac{2 x}{2} = \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 7}{2}$

Divide $x$ by $1$ .

$x = \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Step 6

The final solution is all the values that make $(2 x - 3) (2 x + 7) = 0$ true.

$x = \frac{3}{2}, - \frac{7}{2}$

Step 7

Use each root to create test intervals.

$x < - \frac{7}{2}$

$- \frac{7}{2} < x < \frac{3}{2}$

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

Step 8

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 $x < - \frac{7}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x < - \frac{7}{2}$ and see if this value makes the original inequality true.

$x = - 6$

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

$4 {(- 6)}^{2} + 8 (- 6) - 21 \geq 0$

The left side $75$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- \frac{7}{2} < x < \frac{3}{2}$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{7}{2} < x < \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 0$

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

$4 {(0)}^{2} + 8 (0) - 21 \geq 0$

The left side $- 21$ is less than the right side $0$ , which means that the given statement is false.

False

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.

$4 {(4)}^{2} + 8 (4) - 21 \geq 0$

The left side $75$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

$x < - \frac{7}{2}$ True

$- \frac{7}{2} < x < \frac{3}{2}$ False

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

$x < - \frac{7}{2}$ True

$- \frac{7}{2} < x < \frac{3}{2}$ False

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

Step 9

The solution consists of all of the true intervals.

$x \leq - \frac{7}{2}$ or $x \geq \frac{3}{2}$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{7}{2} or x \geq \frac{3}{2}$

Interval Notation:

$(- \infty, - \frac{7}{2}] \cup [\frac{3}{2}, \infty)$

Step 11

Solve the Inequality for x 4x^2+8x-21>=0

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