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

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

Step 1

Add $x$ to both sides of the inequality.

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

Step 2

Convert the inequality to an equation.

$- 4 x^{2} + 7 + x = 2$

Step 3

Subtract $2$ from both sides of the equation.

$- 4 x^{2} + 7 + x - 2 = 0$

Step 4

Subtract $2$ from $7$ .

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

Step 5

Factor the left side of the equation.

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

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

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

Factor $- 1$ out of $x$ .

$- (4 x^{2}) - 1 (- x) + 5 = 0$

Rewrite $5$ as $- 1 (- 5)$ .

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

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

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

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

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

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 $a \cdot c = 4 \cdot - 5 = - 20$ and whose sum is $b = - 1$ .

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

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

Rewrite $- 1$ as $4$ plus $- 5$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$- (4 x (x + 1) - 5 (x + 1)) = 0$

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

$- ((x + 1) (4 x - 5)) = 0$

Remove unnecessary parentheses.

$- (x + 1) (4 x - 5) = 0$

Step 6

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

$x + 1 = 0$

$4 x - 5 = 0$

Step 7

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 8

Set $4 x - 5$ equal to $0$ and solve for $x$ .

Set $4 x - 5$ equal to $0$ .

$4 x - 5 = 0$

Solve $4 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$4 x = 5$

Divide each term in $4 x = 5$ by $4$ and simplify.

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Divide $x$ by $1$ .

$x = \frac{5}{4}$

Step 9

The final solution is all the values that make $- (x + 1) (4 x - 5) = 0$ true.

$x = - 1, \frac{5}{4}$

Step 10

Use each root to create test intervals.

$x < - 1$

$- 1 < x < \frac{5}{4}$

$x > \frac{5}{4}$

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 $x < - 1$ to see if it makes the inequality true.

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

$x = - 4$

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

$- 4 {(- 4)}^{2} + 7 < - (- 4) + 2$

The left side $- 57$ is less than the right side $6$ , which means that the given statement is always true.

True

Test a value on the interval $- 1 < x < \frac{5}{4}$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < \frac{5}{4}$ and see if this value makes the original inequality true.

$x = 0$

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

$- 4 {(0)}^{2} + 7 < - (0) + 2$

The left side $7$ is not less than the right side $2$ , which means that the given statement is false.

False

Test a value on the interval $x > \frac{5}{4}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{5}{4}$ and see if this value makes the original inequality true.

$x = 4$

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

$- 4 {(4)}^{2} + 7 < - (4) + 2$

The left side $- 57$ is less than the right side $- 2$ , which means that the given statement is always true.

True

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

$x < - 1$ True

$- 1 < x < \frac{5}{4}$ False

$x > \frac{5}{4}$ True

$x < - 1$ True

$- 1 < x < \frac{5}{4}$ False

$x > \frac{5}{4}$ True

Step 12

The solution consists of all of the true intervals.

$x < - 1$ or $x > \frac{5}{4}$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$x < - 1 or x > \frac{5}{4}$

Interval Notation:

$(- \infty, - 1) \cup (\frac{5}{4}, \infty)$

Step 14

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

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