Algebra Examples

$(x + 2) (x - 3) > (x - 1) (x + 1)$

Step 1

Simplify $(x + 2) (x - 3)$ .

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Step 1.1

Rewrite.

$0 + 0 + (x + 2) (x - 3) > (x - 1) (x + 1)$

Step 1.2

Simplify by adding zeros.

$(x + 2) (x - 3) > (x - 1) (x + 1)$

Step 1.3

Expand $(x + 2) (x - 3)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$x (x - 3) + 2 (x - 3) > (x - 1) (x + 1)$

Step 1.3.2

Apply the distributive property.

$x \cdot x + x \cdot - 3 + 2 (x - 3) > (x - 1) (x + 1)$

Step 1.3.3

Apply the distributive property.

$x \cdot x + x \cdot - 3 + 2 x + 2 \cdot - 3 > (x - 1) (x + 1)$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 + 2 x + 2 \cdot - 3 > (x - 1) (x + 1)$

Step 1.4.1.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x + 2 x + 2 \cdot - 3 > (x - 1) (x + 1)$

Step 1.4.1.3

Multiply $2$ by $- 3$ .

$x^{2} - 3 x + 2 x - 6 > (x - 1) (x + 1)$

Step 1.4.2

Add $- 3 x$ and $2 x$ .

$x^{2} - x - 6 > (x - 1) (x + 1)$

Step 2

Simplify $(x - 1) (x + 1)$ .

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Step 2.1

Expand $(x - 1) (x + 1)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$x^{2} - x - 6 > x (x + 1) - 1 (x + 1)$

Step 2.1.2

Apply the distributive property.

$x^{2} - x - 6 > x \cdot x + x \cdot 1 - 1 (x + 1)$

Step 2.1.3

Apply the distributive property.

$x^{2} - x - 6 > x \cdot x + x \cdot 1 - 1 x - 1 \cdot 1$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine the opposite terms in $x \cdot x + x \cdot 1 - 1 x - 1 \cdot 1$ .

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Step 2.2.1.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

$x^{2} - x - 6 > x \cdot x + 1 x - 1 x - 1 \cdot 1$

Step 2.2.1.2

Subtract $1 x$ from $1 x$ .

$x^{2} - x - 6 > x \cdot x + 0 - 1 \cdot 1$

Step 2.2.1.3

Add $x \cdot x$ and $0$ .

$x^{2} - x - 6 > x \cdot x - 1 \cdot 1$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Multiply $x$ by $x$ .

$x^{2} - x - 6 > x^{2} - 1 \cdot 1$

Step 2.2.2.2

Multiply $- 1$ by $1$ .

$x^{2} - x - 6 > x^{2} - 1$

Step 3

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

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Step 3.1

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

$x^{2} - x - 6 - x^{2} > - 1$

Step 3.2

Combine the opposite terms in $x^{2} - x - 6 - x^{2}$ .

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Step 3.2.1

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

$- x - 6 + 0 > - 1$

Step 3.2.2

Add $- x - 6$ and $0$ .

$- x - 6 > - 1$

Step 4

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.1

Add $6$ to both sides of the inequality.

$- x > - 1 + 6$

Step 4.2

Add $- 1$ and $6$ .

$- x > 5$

Step 5

Divide each term in $- x > 5$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- x > 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2

Divide $x$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $5$ by $- 1$ .

$x < - 5$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x < - 5$

Interval Notation:

$(- \infty, - 5)$

Step 7