Solve the Inequality for x 2x^2<13x+7

$2 x^{2} < 13 x + 7$

Subtract $13 x$ from both sides of the inequality.

$2 x^{2} - 13 x < 7$

Convert the inequality to an equation.

$2 x^{2} - 13 x = 7$

Subtract $7$ from both sides of the equation.

$2 x^{2} - 13 x - 7 = 0$

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 $a \cdot c = 2 \cdot - 7 = - 14$ and whose sum is $b = - 13$ .

Factor $- 13$ out of $- 13 x$ .

$2 x^{2} - 13 x - 7 = 0$

Rewrite $- 13$ as $1$ plus $- 14$

$2 x^{2} + (1 - 14) x - 7 = 0$

Apply the distributive property.

$2 x^{2} + 1 x - 14 x - 7 = 0$

Multiply $x$ by $1$ .

$2 x^{2} + x - 14 x - 7 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{2} + x) - 14 x - 7 = 0$

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

$x (2 x + 1) - 7 (2 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(2 x + 1) (x - 7) = 0$

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

$2 x + 1 = 0$

$x - 7 = 0$

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

The final solution is all the values that make $(2 x + 1) (x - 7) = 0$ true.

$x = - \frac{1}{2}, 7$

Use each root to create test intervals.

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

$- \frac{1}{2} < x < 7$

$x > 7$

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 $x < - \frac{1}{2}$ to see if it makes the inequality true.

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

$x = - 3$

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

$2 {(- 3)}^{2} < 13 (- 3) + 7$

The left side $18$ is not less than the right side $- 32$ , which means that the given statement is false.

False

Test a value on the interval $- \frac{1}{2} < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{1}{2} < x < 7$ and see if this value makes the original inequality true.

$x = 0$

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

$2 {(0)}^{2} < 13 (0) + 7$

The left side $0$ is less than the right side $7$ , which means that the given statement is always true.

True

Test a value on the interval $x > 7$ to see if it makes the inequality true.

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

$2 {(10)}^{2} < 13 (10) + 7$

The left side $200$ is not less than the right side $137$ , which means that the given statement is false.

False

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

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

$- \frac{1}{2} < x < 7$ True

$x > 7$ False

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

$- \frac{1}{2} < x < 7$ True

$x > 7$ False

The solution consists of all of the true intervals.

$- \frac{1}{2} < x < 7$

The result can be shown in multiple forms.

Inequality Form:

$- \frac{1}{2} < x < 7$

Interval Notation:

$(- \frac{1}{2}, 7)$

Solve the Inequality for x 2x^2<13x+7

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