Solve the Inequality for x 5x^2-25x<x-5

$5 x^{2} - 25 x < x - 5$

Step 1

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

Subtract $x$ from both sides of the inequality.

$5 x^{2} - 25 x - x < - 5$

Subtract $x$ from $- 25 x$ .

$5 x^{2} - 26 x < - 5$

Step 2

Convert the inequality to an equation.

$5 x^{2} - 26 x = - 5$

Step 3

Add $5$ to both sides of the equation.

$5 x^{2} - 26 x + 5 = 0$

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 $a \cdot c = 5 \cdot 5 = 25$ and whose sum is $b = - 26$ .

Factor $- 26$ out of $- 26 x$ .

$5 x^{2} - 26 x + 5 = 0$

Rewrite $- 26$ as $- 1$ plus $- 25$

$5 x^{2} + (- 1 - 25) x + 5 = 0$

Apply the distributive property.

$5 x^{2} - 1 x - 25 x + 5 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(5 x^{2} - 1 x) - 25 x + 5 = 0$

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

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

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

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

Step 5

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

$5 x - 1 = 0$

$x - 5 = 0$

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 $x$ .

Add $1$ to 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 $1$ .

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

Step 7

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 8

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

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

Step 9

Use each root to create test intervals.

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

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

$x > 5$

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 $x < \frac{1}{5}$ 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.

$5 {(0)}^{2} - 25 \cdot 0 < (0) - 5$

The left side $0$ is not less than the right side $- 5$ , which means that the given statement is false.

False

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

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

$x = 3$

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

$5 {(3)}^{2} - 25 \cdot 3 < (3) - 5$

The left side $- 30$ is less than the right side $- 2$ , which means that the given statement is always true.

True

Test a value on the interval $x > 5$ to see if it makes the inequality true.

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

$5 {(8)}^{2} - 25 \cdot 8 < (8) - 5$

The left side $120$ is not less 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.

$x < \frac{1}{5}$ False

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

$x > 5$ False

$x < \frac{1}{5}$ False

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

$x > 5$ False

Step 11

The solution consists of all of the true intervals.

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

Step 12

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(\frac{1}{5}, 5)$

Step 13

Solve the Inequality for x 5x^2-25x<x-5

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