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

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

Step 1

Subtract $5 x$ from both sides of the inequality.

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

Step 2

Convert the inequality to an equation.

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

Step 3

Add $2$ to both sides of the equation.

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

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

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

Rewrite $- 5$ as $- 1$ plus $- 4$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 5

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

$2 x - 1 = 0$

$x - 2 = 0$

Step 6

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 $x$ .

Add $1$ to 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 $1$ .

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

Step 7

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 8

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

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

Step 9

Use each root to create test intervals.

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

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

$x > 2$

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

$x = 0$

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

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

The left side $0$ is greater than the right side $- 2$ , which means that the given statement is false.

False

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

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

$x = 1$

Replace $x$ with $1$ in the original inequality.

$2 {(1)}^{2} \leq 5 (1) - 2$

The left side $2$ is less than the right side $3$ , which means that the given statement is always true.

True

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

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

The left side $32$ is greater than the right side $18$ , 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

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

$x > 2$ False

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

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

$x > 2$ False

Step 11

The solution consists of all of the true intervals.

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

Step 12

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Step 13

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

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