Solve the Inequality for x 5x^2+22x<=6x-3

$5 x^{2} + 22 x \leq 6 x - 3$

Step 1

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

Subtract $6 x$ from both sides of the inequality.

$5 x^{2} + 22 x - 6 x \leq - 3$

Subtract $6 x$ from $22 x$ .

$5 x^{2} + 16 x \leq - 3$

Step 2

Convert the inequality to an equation.

$5 x^{2} + 16 x = - 3$

Step 3

Add $3$ to both sides of the equation.

$5 x^{2} + 16 x + 3 = 0$

Step 4

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 $a \cdot c = 5 \cdot 3 = 15$ and whose sum is $b = 16$ .

Factor $16$ out of $16 x$ .

$5 x^{2} + 16 (x) + 3 = 0$

Rewrite $16$ as $1$ plus $15$

$5 x^{2} + (1 + 15) x + 3 = 0$

Apply the distributive property.

$5 x^{2} + 1 x + 15 x + 3 = 0$

Multiply $x$ by $1$ .

$5 x^{2} + x + 15 x + 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(5 x^{2} + x) + 15 x + 3 = 0$

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

$x (5 x + 1) + 3 (5 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 x + 1$ .

$(5 x + 1) (x + 3) = 0$

Step 5

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

$5 x + 1 = 0$

$x + 3 = 0$

Step 6

Set $5 x + 1$ equal to $0$ and solve for $x$ .

Set $5 x + 1$ equal to $0$ .

$5 x + 1 = 0$

Solve $5 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$5 x = - 1$

Divide each term in $5 x = - 1$ by $5$ and simplify.

Divide each term in $5 x = - 1$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 7

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 8

The final solution is all the values that make $(5 x + 1) (x + 3) = 0$ true.

$x = - \frac{1}{5}, - 3$

Step 9

Use each root to create test intervals.

$x < - 3$

$- 3 < x < - \frac{1}{5}$

$x > - \frac{1}{5}$

Step 10

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 $x < - 3$ to see if it makes the inequality true.

Choose a value on the interval $x < - 3$ and see if this value makes the original inequality true.

$x = - 6$

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

$5 {(- 6)}^{2} + 22 (- 6) \leq 6 (- 6) - 3$

The left side $48$ is greater than the right side $- 39$ , which means that the given statement is false.

False

Test a value on the interval $- 3 < x < - \frac{1}{5}$ to see if it makes the inequality true.

Choose a value on the interval $- 3 < x < - \frac{1}{5}$ and see if this value makes the original inequality true.

$x = - 2$

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

$5 {(- 2)}^{2} + 22 (- 2) \leq 6 (- 2) - 3$

The left side $- 24$ is less than the right side $- 15$ , which means that the given statement is always true.

True

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

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

$x = 0$

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

$5 {(0)}^{2} + 22 (0) \leq 6 (0) - 3$

The left side $0$ is greater than the right side $- 3$ , which means that the given statement is false.

False

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

$x < - 3$ False

$- 3 < x < - \frac{1}{5}$ True

$x > - \frac{1}{5}$ False

$x < - 3$ False

$- 3 < x < - \frac{1}{5}$ True

$x > - \frac{1}{5}$ False

Step 11

The solution consists of all of the true intervals.

$- 3 \leq x \leq - \frac{1}{5}$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- 3 \leq x \leq - \frac{1}{5}$

Interval Notation:

$[- 3, - \frac{1}{5}]$

Step 13

Solve the Inequality for x 5x^2+22x<=6x-3

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