Solve the Inequality for x -4x^2+7>=5x+1

$- 4 x^{2} + 7 \geq 5 x + 1$

Step 1

Subtract $5 x$ from both sides of the inequality.

$- 4 x^{2} + 7 - 5 x \geq 1$

Step 2

Convert the inequality to an equation.

$- 4 x^{2} + 7 - 5 x = 1$

Step 3

Subtract $1$ from both sides of the equation.

$- 4 x^{2} + 7 - 5 x - 1 = 0$

Step 4

Subtract $1$ from $7$ .

$- 4 x^{2} - 5 x + 6 = 0$

Step 5

Factor the left side of the equation.

Factor $- 1$ out of $- 4 x^{2} - 5 x + 6$ .

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

$- (4 x^{2}) - 5 x + 6 = 0$

Factor $- 1$ out of $- 5 x$ .

$- (4 x^{2}) - (5 x) + 6 = 0$

Rewrite $6$ as $- 1 (- 6)$ .

$- (4 x^{2}) - (5 x) - 1 \cdot - 6 = 0$

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

$- (4 x^{2} + 5 x) - 1 \cdot - 6 = 0$

Factor $- 1$ out of $- (4 x^{2} + 5 x) - 1 (- 6)$ .

$- (4 x^{2} + 5 x - 6) = 0$

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 $a \cdot c = 4 \cdot - 6 = - 24$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$- (4 x^{2} + 5 (x) - 6) = 0$

Rewrite $5$ as $- 3$ plus $8$

$- (4 x^{2} + (- 3 + 8) x - 6) = 0$

Apply the distributive property.

$- (4 x^{2} - 3 x + 8 x - 6) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- ((4 x^{2} - 3 x) + 8 x - 6) = 0$

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

$- (x (4 x - 3) + 2 (4 x - 3)) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x - 3$ .

$- ((4 x - 3) (x + 2)) = 0$

Remove unnecessary parentheses.

$- (4 x - 3) (x + 2) = 0$

Step 6

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

$4 x - 3 = 0$

$x + 2 = 0$

Step 7

Set $4 x - 3$ equal to $0$ and solve for $x$ .

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Solve $4 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$4 x = 3$

Divide each term in $4 x = 3$ by $4$ and simplify.

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 8

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 9

The final solution is all the values that make $- (4 x - 3) (x + 2) = 0$ true.

$x = \frac{3}{4}, - 2$

Step 10

Use each root to create test intervals.

$x < - 2$

$- 2 < x < \frac{3}{4}$

$x > \frac{3}{4}$

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 $x < - 2$ 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.

$- 4 {(- 4)}^{2} + 7 \geq 5 (- 4) + 1$

The left side $- 57$ is less than the right side $- 19$ , which means that the given statement is false.

False

Test a value on the interval $- 2 < x < \frac{3}{4}$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < \frac{3}{4}$ and see if this value makes the original inequality true.

$x = 0$

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

$- 4 {(0)}^{2} + 7 \geq 5 (0) + 1$

The left side $7$ is greater than the right side $1$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{3}{4}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{3}{4}$ and see if this value makes the original inequality true.

$x = 3$

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

$- 4 {(3)}^{2} + 7 \geq 5 (3) + 1$

The left side $- 29$ is less than the right side $16$ , which means that the given statement is false.

False

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

$x < - 2$ False

$- 2 < x < \frac{3}{4}$ True

$x > \frac{3}{4}$ False

$x < - 2$ False

$- 2 < x < \frac{3}{4}$ True

$x > \frac{3}{4}$ False

Step 12

The solution consists of all of the true intervals.

$- 2 \leq x \leq \frac{3}{4}$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq x \leq \frac{3}{4}$

Interval Notation:

$[- 2, \frac{3}{4}]$

Step 14

Solve the Inequality for x -4x^2+7>=5x+1

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