Solve the Inequality for x 5x^2-10<=2x-7

$5 x^{2} - 10 \leq 2 x - 7$

Step 1

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

$5 x^{2} - 10 - 2 x \leq - 7$

Step 2

Convert the inequality to an equation.

$5 x^{2} - 10 - 2 x = - 7$

Step 3

Add $7$ to both sides of the equation.

$5 x^{2} - 10 - 2 x + 7 = 0$

Step 4

Add $- 10$ and $7$ .

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

Step 5

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 $b = - 2$ .

Factor $- 2$ out of $- 2 x$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $5 x + 3$ .

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

Step 6

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

$5 x + 3 = 0$

$x - 1 = 0$

Step 7

Set $5 x + 3$ equal to $0$ and solve for $x$ .

Set $5 x + 3$ equal to $0$ .

$5 x + 3 = 0$

Solve $5 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$5 x = - 3$

Divide each term in $5 x = - 3$ by $5$ and simplify.

Divide each term in $5 x = - 3$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 8

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 9

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

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

Step 10

Use each root to create test intervals.

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

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

$x > 1$

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 $x < - \frac{3}{5}$ to see if it makes the inequality true.

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

$x = - 3$

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

$5 {(- 3)}^{2} - 10 \leq 2 (- 3) - 7$

The left side $35$ is greater than the right side $- 13$ , which means that the given statement is false.

False

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

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

$x = 0$

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

$5 {(0)}^{2} - 10 \leq 2 (0) - 7$

The left side $- 10$ is less than the right side $- 7$ , 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.

$5 {(4)}^{2} - 10 \leq 2 (4) - 7$

The left side $70$ is greater than the right side $1$ , which means that the given statement is false.

False

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

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

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

$x > 1$ False

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

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

$x > 1$ False

Step 12

The solution consists of all of the true intervals.

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

Step 13

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Step 14

Solve the Inequality for x 5x^2-10<=2x-7

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