Algebra Examples

$(x + 2) (x - 1) + 26 < (x + 4) (x + 5)$

Step 1

Simplify $(x + 2) (x - 1) + 26$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Expand $(x + 2) (x - 1)$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

$x (x - 1) + 2 (x - 1) + 26 < (x + 4) (x + 5)$

Step 1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 2 (x - 1) + 26 < (x + 4) (x + 5)$

Step 1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 2 x + 2 \cdot - 1 + 26 < (x + 4) (x + 5)$

Step 1.1.2

Simplify and combine like terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 + 2 x + 2 \cdot - 1 + 26 < (x + 4) (x + 5)$

Step 1.1.2.1.2

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x + 2 x + 2 \cdot - 1 + 26 < (x + 4) (x + 5)$

Step 1.1.2.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x + 2 x + 2 \cdot - 1 + 26 < (x + 4) (x + 5)$

Step 1.1.2.1.4

Multiply $2$ by $- 1$ .

$x^{2} - x + 2 x - 2 + 26 < (x + 4) (x + 5)$

Step 1.1.2.2

Add $- x$ and $2 x$ .

$x^{2} + x - 2 + 26 < (x + 4) (x + 5)$

Step 1.2

Add $- 2$ and $26$ .

$x^{2} + x + 24 < (x + 4) (x + 5)$

Step 2

Simplify $(x + 4) (x + 5)$ .

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Step 2.1

Expand $(x + 4) (x + 5)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$x^{2} + x + 24 < x (x + 5) + 4 (x + 5)$

Step 2.1.2

Apply the distributive property.

$x^{2} + x + 24 < x \cdot x + x \cdot 5 + 4 (x + 5)$

Step 2.1.3

Apply the distributive property.

$x^{2} + x + 24 < x \cdot x + x \cdot 5 + 4 x + 4 \cdot 5$

Step 2.2

Simplify and combine like terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x + 24 < x^{2} + x \cdot 5 + 4 x + 4 \cdot 5$

Step 2.2.1.2

Move $5$ to the left of $x$ .

$x^{2} + x + 24 < x^{2} + 5 \cdot x + 4 x + 4 \cdot 5$

Step 2.2.1.3

Multiply $4$ by $5$ .

$x^{2} + x + 24 < x^{2} + 5 x + 4 x + 20$

Step 2.2.2

Add $5 x$ and $4 x$ .

$x^{2} + x + 24 < x^{2} + 9 x + 20$

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.

$x^{2} + x + 24 - x^{2} < 9 x + 20$

Step 3.2

Subtract $9 x$ from both sides of the inequality.

$x^{2} + x + 24 - x^{2} - 9 x < 20$

Step 3.3

Combine the opposite terms in $x^{2} + x + 24 - x^{2} - 9 x$ .

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Step 3.3.1

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

$x + 24 + 0 - 9 x < 20$

Step 3.3.2

Add $x + 24$ and $0$ .

$x + 24 - 9 x < 20$

Step 3.4

Subtract $9 x$ from $x$ .

$- 8 x + 24 < 20$

Step 4

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

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Step 4.1

Subtract $24$ from both sides of the inequality.

$- 8 x < 20 - 24$

Step 4.2

Subtract $24$ from $20$ .

$- 8 x < - 4$

Step 5

Divide each term in $- 8 x < - 4$ by $- 8$ and simplify.

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Step 5.1

Divide each term in $- 8 x < - 4$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} > \frac{- 4}{- 8}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 8$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} > \frac{- 4}{- 8}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 4}{- 8}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $- 4$ and $- 8$ .

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Step 5.3.1.1

Factor $- 4$ out of $- 4$ .

$x > \frac{- 4 (1)}{- 8}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $- 4$ out of $- 8$ .

$x > \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 5.3.1.2.2

Cancel the common factor.

$x > \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 5.3.1.2.3

Rewrite the expression.

$x > \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{1}{2}$

Interval Notation:

$(\frac{1}{2}, \infty)$

Step 7