Pre-Algebra Examples

$15 (x - 2) - x (1 - x) > x^{2}$

Step 1

Simplify $15 (x - 2) - x (1 - x)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$15 x + 15 \cdot - 2 - x (1 - x) > x^{2}$

Step 1.1.2

Multiply $15$ by $- 2$ .

$15 x - 30 - x (1 - x) > x^{2}$

Step 1.1.3

Apply the distributive property.

$15 x - 30 - x \cdot 1 - x (- x) > x^{2}$

Step 1.1.4

Multiply $- 1$ by $1$ .

$15 x - 30 - x - x (- x) > x^{2}$

Step 1.1.5

Rewrite using the commutative property of multiplication.

$15 x - 30 - x - 1 \cdot - 1 x \cdot x > x^{2}$

Step 1.1.6

Simplify each term.

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Step 1.1.6.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.6.1.1

Move $x$ .

$15 x - 30 - x - 1 \cdot - 1 (x \cdot x) > x^{2}$

Step 1.1.6.1.2

Multiply $x$ by $x$ .

$15 x - 30 - x - 1 \cdot - 1 x^{2} > x^{2}$

Step 1.1.6.2

Multiply $- 1$ by $- 1$ .

$15 x - 30 - x + 1 x^{2} > x^{2}$

Step 1.1.6.3

Multiply $x^{2}$ by $1$ .

$15 x - 30 - x + x^{2} > x^{2}$

Step 1.2

Subtract $x$ from $15 x$ .

$14 x - 30 + x^{2} > x^{2}$

Step 2

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

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Step 2.1

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

$14 x - 30 + x^{2} - x^{2} > 0$

Step 2.2

Combine the opposite terms in $14 x - 30 + x^{2} - x^{2}$ .

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Step 2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$14 x - 30 + 0 > 0$

Step 2.2.2

Add $14 x - 30$ and $0$ .

$14 x - 30 > 0$

Step 3

Add $30$ to both sides of the inequality.

$14 x > 30$

Step 4

Divide each term in $14 x > 30$ by $14$ and simplify.

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Step 4.1

Divide each term in $14 x > 30$ by $14$ .

$\frac{14 x}{14} > \frac{30}{14}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $14$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{14 x}{14} > \frac{30}{14}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x > \frac{30}{14}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $30$ and $14$ .

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Step 4.3.1.1

Factor $2$ out of $30$ .

$x > \frac{2 (15)}{14}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $14$ .

$x > \frac{2 \cdot 15}{2 \cdot 7}$

Step 4.3.1.2.2

Cancel the common factor.

$x > \frac{2 \cdot 15}{2 \cdot 7}$

Step 4.3.1.2.3

Rewrite the expression.

$x > \frac{15}{7}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{15}{7}$

Interval Notation:

$(\frac{15}{7}, \infty)$

Step 6