Algebra Examples

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

Step 1

Simplify $\frac{7}{2} x$ .

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Step 1.1

Rewrite.

$x^{2} - 2 < 0 + 0 + \frac{7}{2} x$

Step 1.2

Simplify by adding zeros.

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

Step 1.3

Combine $\frac{7}{2}$ and $x$ .

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

Step 2

Subtract $\frac{7 x}{2}$ from both sides of the inequality.

$x^{2} - 2 - \frac{7 x}{2} < 0$

Step 3

Multiply through by the least common denominator $2$ , then simplify.

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Step 3.1

Apply the distributive property.

$2 x^{2} + 2 \cdot - 2 + 2 (- \frac{7 x}{2}) < 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $2$ by $- 2$ .

$2 x^{2} - 4 + 2 (- \frac{7 x}{2}) < 0$

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

Move the leading negative in $- \frac{7 x}{2}$ into the numerator.

$2 x^{2} - 4 + 2 (\frac{- 7 x}{2}) < 0$

Step 3.2.2.2

Cancel the common factor.

$2 x^{2} - 4 + 2 (\frac{- 7 x}{2}) < 0$

Step 3.2.2.3

Rewrite the expression.

$2 x^{2} - 4 - 7 x < 0$

Step 3.3

Move $- 4$ .

$2 x^{2} - 7 x - 4 < 0$

Step 4

Convert the inequality to an equation.

$2 x^{2} - 7 x - 4 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 2$ , $b = - 7$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (2 \cdot - 4)}}{2 \cdot 2}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 2 \cdot - 4}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot - 4$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{7 \pm \sqrt{49 - 8 \cdot - 4}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $- 4$ .

$x = \frac{7 \pm \sqrt{49 + 32}}{2 \cdot 2}$

Step 7.1.3

Add $49$ and $32$ .

$x = \frac{7 \pm \sqrt{81}}{2 \cdot 2}$

Step 7.1.4

Rewrite $81$ as $9^{2}$ .

$x = \frac{7 \pm \sqrt{9^{2}}}{2 \cdot 2}$

Step 7.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{7 \pm 9}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$x = \frac{7 \pm 9}{4}$

Step 8

Consolidate the solutions.

$x = 4, - \frac{1}{2}$

Step 9

Use each root to create test intervals.

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

$- \frac{1}{2} < x < 4$

$x > 4$

Step 10

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 10.1

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

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Step 10.1.1

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

$x = - 3$

Step 10.1.2

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

${(- 3)}^{2} - 2 < \frac{7}{2} \cdot (- 3)$

Step 10.1.3

The left side $7$ is not less than the right side $- 10.5$ , which means that the given statement is false.

False

Step 10.2

Test a value on the interval $- \frac{1}{2} < x < 4$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $- \frac{1}{2} < x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 10.2.2

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

${(0)}^{2} - 2 < \frac{7}{2} \cdot (0)$

Step 10.2.3

The left side $- 2$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 10.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 10.3.2

Replace $x$ with $6$ in the original inequality.

${(6)}^{2} - 2 < \frac{7}{2} \cdot (6)$

Step 10.3.3

The left side $34$ is not less than the right side $21$ , which means that the given statement is false.

False

Step 10.4

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

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

$- \frac{1}{2} < x < 4$ True

$x > 4$ False

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

$- \frac{1}{2} < x < 4$ True

$x > 4$ False

Step 11

The solution consists of all of the true intervals.

$- \frac{1}{2} < x < 4$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- \frac{1}{2} < x < 4$

Interval Notation:

$(- \frac{1}{2}, 4)$

Step 13