Solve the Inequality for x x^2<10x-3

$x^{2} < 10 x - 3$

Step 1

Subtract $10 x$ from both sides of the inequality.

$x^{2} - 10 x < - 3$

Step 2

Add $3$ to both sides of the inequality.

$x^{2} - 10 x + 3 < 0$

Step 3

Convert the inequality to an equation.

$x^{2} - 10 x + 3 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 1$ , $b = - 10$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 6

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 3$ .

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3}}{2 \cdot 1}$

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12}}{2 \cdot 1}$

Subtract $12$ from $100$ .

$x = \frac{10 \pm \sqrt{88}}{2 \cdot 1}$

Rewrite $88$ as $2^{2} \cdot 22$ .

Factor $4$ out of $88$ .

$x = \frac{10 \pm \sqrt{4 (22)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 22}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{22}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 2 \sqrt{22}}{2}$

Simplify $\frac{10 \pm 2 \sqrt{22}}{2}$ .

$x = 5 \pm \sqrt{22}$

Step 7

Consolidate the solutions.

$x = 5 + \sqrt{22}, 5 - \sqrt{22}$

Step 8

Use each root to create test intervals.

$x < 5 - \sqrt{22}$

$5 - \sqrt{22} < x < 5 + \sqrt{22}$

$x > 5 + \sqrt{22}$

Step 9

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 $x < 5 - \sqrt{22}$ to see if it makes the inequality true.

Choose a value on the interval $x < 5 - \sqrt{22}$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} < 10 (0) - 3$

The left side $0$ is not less than the right side $- 3$ , which means that the given statement is false.

False

Test a value on the interval $5 - \sqrt{22} < x < 5 + \sqrt{22}$ to see if it makes the inequality true.

Choose a value on the interval $5 - \sqrt{22} < x < 5 + \sqrt{22}$ and see if this value makes the original inequality true.

$x = 5$

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

${(5)}^{2} < 10 (5) - 3$

The left side $25$ is less than the right side $47$ , which means that the given statement is always true.

True

Test a value on the interval $x > 5 + \sqrt{22}$ to see if it makes the inequality true.

Choose a value on the interval $x > 5 + \sqrt{22}$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

${(12)}^{2} < 10 (12) - 3$

The left side $144$ is not less than the right side $117$ , which means that the given statement is false.

False

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

$x < 5 - \sqrt{22}$ False

$5 - \sqrt{22} < x < 5 + \sqrt{22}$ True

$x > 5 + \sqrt{22}$ False

$x < 5 - \sqrt{22}$ False

$5 - \sqrt{22} < x < 5 + \sqrt{22}$ True

$x > 5 + \sqrt{22}$ False

Step 10

The solution consists of all of the true intervals.

$5 - \sqrt{22} < x < 5 + \sqrt{22}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$5 - \sqrt{22} < x < 5 + \sqrt{22}$

Interval Notation:

$(5 - \sqrt{22}, 5 + \sqrt{22})$

Step 12

Solve the Inequality for x x^2<10x-3

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