Algebra Examples

$\frac{1}{2} x - 7 \leq x^{2}$

Step 1

Combine $\frac{1}{2}$ and $x$ .

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

Step 2

Subtract $x^{2}$ from both sides of the inequality.

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

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

Step 3.2.2

Multiply $2$ by $- 7$ .

$x - 14 + 2 (- x^{2}) \leq 0$

Step 3.2.3

Multiply $- 1$ by $2$ .

$x - 14 - 2 x^{2} \leq 0$

Step 3.3

Move $- 14$ .

$x - 2 x^{2} - 14 \leq 0$

Step 3.4

Reorder $x$ and $- 2 x^{2}$ .

$- 2 x^{2} + x - 14 \leq 0$

Step 4

Convert the inequality to an equation.

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

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 $a = - 2$ , $b = 1$ , and $c = - 14$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 2 \cdot - 14)}}{2 \cdot - 2}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 2 \cdot - 14}}{2 \cdot - 2}$

Step 7.1.2

Multiply $- 4 \cdot - 2 \cdot - 14$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 1 \pm \sqrt{1 + 8 \cdot - 14}}{2 \cdot - 2}$

Step 7.1.2.2

Multiply $8$ by $- 14$ .

$x = \frac{- 1 \pm \sqrt{1 - 112}}{2 \cdot - 2}$

Step 7.1.3

Subtract $112$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 111}}{2 \cdot - 2}$

Step 7.1.4

Rewrite $- 111$ as $- 1 (111)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 111}}{2 \cdot - 2}$

Step 7.1.5

Rewrite $\sqrt{- 1 (111)}$ as $\sqrt{- 1} \cdot \sqrt{111}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{111}}{2 \cdot - 2}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{111}}{2 \cdot - 2}$

Step 7.2

Multiply $2$ by $- 2$ .

$x = \frac{- 1 \pm i \sqrt{111}}{- 4}$

Step 7.3

Simplify $\frac{- 1 \pm i \sqrt{111}}{- 4}$ .

$x = \frac{1 \pm i \sqrt{111}}{4}$

Step 8

Identify the leading coefficient.

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Step 8.1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 8.1.1

Move $- 7$ .

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

Step 8.1.2

Reorder $\frac{x}{2}$ and $- x^{2}$ .

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

Step 8.2

The leading term in a polynomial is the term with the highest degree.

$- x^{2}$

Step 8.3

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 1$

Step 9

Since there are no real x-intercepts and the leading coefficient is negative, the parabola opens down and $\frac{x}{2} - 7 - x^{2}$ is always less than $0$ .

All real numbers

Step 10

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$