Solve the Inequality for x 4x^2+14x+14>=-3x+10

$4 x^{2} + 14 x + 14 \geq - 3 x + 10$

Move all terms containing $x$ to the left side of the inequality.

Add $3 x$ to both sides of the inequality.

$4 x^{2} + 14 x + 14 + 3 x \geq 10$

Add $14 x$ and $3 x$ .

$4 x^{2} + 17 x + 14 \geq 10$

Convert the inequality to an equation.

$4 x^{2} + 17 x + 14 = 10$

Subtract $10$ from both sides of the equation.

$4 x^{2} + 17 x + 14 - 10 = 0$

Subtract $10$ from $14$ .

$4 x^{2} + 17 x + 4 = 0$

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 $a \cdot c = 4 \cdot 4 = 16$ and whose sum is $b = 17$ .

Factor $17$ out of $17 x$ .

$4 x^{2} + 17 (x) + 4 = 0$

Rewrite $17$ as $1$ plus $16$

$4 x^{2} + (1 + 16) x + 4 = 0$

Apply the distributive property.

$4 x^{2} + 1 x + 16 x + 4 = 0$

Multiply $x$ by $1$ .

$4 x^{2} + x + 16 x + 4 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} + x) + 16 x + 4 = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

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

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

$4 x + 1 = 0$

$x + 4 = 0$

Set $4 x + 1$ equal to $0$ and solve for $x$ .

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Solve $4 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Divide each term in $4 x = - 1$ by $4$ and simplify.

Divide each term in $4 x = - 1$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

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

Use each root to create test intervals.

$x < - 4$

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

$x > - \frac{1}{4}$

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 $x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

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

$4 {(- 6)}^{2} + 14 (- 6) + 14 \geq - 3 \cdot - 6 + 10$

The left side $74$ is greater than the right side $28$ , which means that the given statement is always true.

True

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

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

$x = - 2$

Replace $x$ with $- 2$ in the original inequality.

$4 {(- 2)}^{2} + 14 (- 2) + 14 \geq - 3 \cdot - 2 + 10$

The left side $2$ is less than the right side $16$ , which means that the given statement is false.

False

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

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

$x = 0$

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

$4 {(0)}^{2} + 14 (0) + 14 \geq - 3 \cdot 0 + 10$

The left side $14$ is greater than the right side $10$ , which means that the given statement is always true.

True

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

$x < - 4$ True

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

$x > - \frac{1}{4}$ True

$x < - 4$ True

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

$x > - \frac{1}{4}$ True

The solution consists of all of the true intervals.

$x \leq - 4$ or $x \geq - \frac{1}{4}$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 4 or x \geq - \frac{1}{4}$

Interval Notation:

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

Solve the Inequality for x 4x^2+14x+14>=-3x+10

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