Solve the Inequality for x 2x^2+5x<=2x+5

$2 x^{2} + 5 x \leq 2 x + 5$

Step 1

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

Subtract $2 x$ from both sides of the inequality.

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

Subtract $2 x$ from $5 x$ .

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

Step 2

Convert the inequality to an equation.

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

Step 3

Subtract $5$ from both sides of the equation.

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

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 $a \cdot c = 2 \cdot - 5 = - 10$ and whose sum is $b = 3$ .

Factor $3$ out of $3 x$ .

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

Rewrite $3$ as $- 2$ plus $5$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$2 x (x - 1) + 5 (x - 1) = 0$

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

$(x - 1) (2 x + 5) = 0$

Step 5

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

$x - 1 = 0$

$2 x + 5 = 0$

Step 6

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 7

Set $2 x + 5$ equal to $0$ and solve for $x$ .

Set $2 x + 5$ equal to $0$ .

$2 x + 5 = 0$

Solve $2 x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Divide each term in $2 x = - 5$ by $2$ and simplify.

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 8

The final solution is all the values that make $(x - 1) (2 x + 5) = 0$ true.

$x = 1, - \frac{5}{2}$

Step 9

Use each root to create test intervals.

$x < - \frac{5}{2}$

$- \frac{5}{2} < x < 1$

$x > 1$

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 $x < - \frac{5}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x < - \frac{5}{2}$ and see if this value makes the original inequality true.

$x = - 5$

Replace $x$ with $- 5$ in the original inequality.

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

The left side $25$ is greater than the right side $- 5$ , which means that the given statement is false.

False

Test a value on the interval $- \frac{5}{2} < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{5}{2} < x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

$2 {(0)}^{2} + 5 (0) \leq 2 (0) + 5$

The left side $0$ is less than the right side $5$ , which means that the given statement is always true.

True

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.

$2 {(4)}^{2} + 5 (4) \leq 2 (4) + 5$

The left side $52$ is greater than the right side $13$ , which means that the given statement is false.

False

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

$x < - \frac{5}{2}$ False

$- \frac{5}{2} < x < 1$ True

$x > 1$ False

$x < - \frac{5}{2}$ False

$- \frac{5}{2} < x < 1$ True

$x > 1$ False

Step 11

The solution consists of all of the true intervals.

$- \frac{5}{2} \leq x \leq 1$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- \frac{5}{2} \leq x \leq 1$

Interval Notation:

$[- \frac{5}{2}, 1]$

Step 13

Solve the Inequality for x 2x^2+5x<=2x+5

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