Algebra Examples

${(2 x - 5)}^{2} - 0.5 x < (2 x - 1) (2 x + 1) - 15$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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Step 1.1.1.1

Apply the distributive property.

${(2 x - 5)}^{2} - 0.5 x < 2 x (2 x + 1) - 1 (2 x + 1) - 15$

Step 1.1.1.2

Apply the distributive property.

${(2 x - 5)}^{2} - 0.5 x < 2 x (2 x) + 2 x \cdot 1 - 1 (2 x + 1) - 15$

Step 1.1.1.3

Apply the distributive property.

${(2 x - 5)}^{2} - 0.5 x < 2 x (2 x) + 2 x \cdot 1 - 1 (2 x) - 1 \cdot 1 - 15$

Step 1.1.2

Combine the opposite terms in $2 x (2 x) + 2 x \cdot 1 - 1 (2 x) - 1 \cdot 1$ .

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Step 1.1.2.1

Reorder the factors in the terms $2 x \cdot 1$ and $- 1 (2 x)$ .

${(2 x - 5)}^{2} - 0.5 x < 2 x (2 x) + 1 \cdot 2 x - 1 \cdot 2 x - 1 \cdot 1 - 15$

Step 1.1.2.2

Subtract $2 x$ from $1 \cdot 2 x$ .

${(2 x - 5)}^{2} - 0.5 x < 2 x (2 x) + 0 - 1 \cdot 1 - 15$

Step 1.1.2.3

Add $2 x (2 x)$ and $0$ .

${(2 x - 5)}^{2} - 0.5 x < 2 x (2 x) - 1 \cdot 1 - 15$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

Rewrite using the commutative property of multiplication.

${(2 x - 5)}^{2} - 0.5 x < 2 \cdot 2 x \cdot x - 1 \cdot 1 - 15$

Step 1.1.3.2

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

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Step 1.1.3.2.1

Move $x$ .

${(2 x - 5)}^{2} - 0.5 x < 2 \cdot 2 (x \cdot x) - 1 \cdot 1 - 15$

Step 1.1.3.2.2

Multiply $x$ by $x$ .

${(2 x - 5)}^{2} - 0.5 x < 2 \cdot 2 x^{2} - 1 \cdot 1 - 15$

Step 1.1.3.3

Multiply $2$ by $2$ .

${(2 x - 5)}^{2} - 0.5 x < 4 x^{2} - 1 \cdot 1 - 15$

Step 1.1.3.4

Multiply $- 1$ by $1$ .

${(2 x - 5)}^{2} - 0.5 x < 4 x^{2} - 1 - 15$

Step 1.2

Subtract $15$ from $- 1$ .

${(2 x - 5)}^{2} - 0.5 x < 4 x^{2} - 16$

Step 2

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

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Step 2.1

Subtract $4 x^{2}$ from both sides of the inequality.

${(2 x - 5)}^{2} - 0.5 x - 4 x^{2} < - 16$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite ${(2 x - 5)}^{2}$ as $(2 x - 5) (2 x - 5)$ .

$(2 x - 5) (2 x - 5) - 0.5 x - 4 x^{2} < - 16$

Step 2.2.2

Expand $(2 x - 5) (2 x - 5)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$2 x (2 x - 5) - 5 (2 x - 5) - 0.5 x - 4 x^{2} < - 16$

Step 2.2.2.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5) - 0.5 x - 4 x^{2} < - 16$

Step 2.2.2.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - 0.5 x - 4 x^{2} < - 16$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - 0.5 x - 4 x^{2} < - 16$

Step 2.2.3.1.2

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

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Step 2.2.3.1.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - 0.5 x - 4 x^{2} < - 16$

Step 2.2.3.1.2.2

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - 0.5 x - 4 x^{2} < - 16$

Step 2.2.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - 0.5 x - 4 x^{2} < - 16$

Step 2.2.3.1.4

Multiply $- 5$ by $2$ .

$4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5 - 0.5 x - 4 x^{2} < - 16$

Step 2.2.3.1.5

Multiply $2$ by $- 5$ .

$4 x^{2} - 10 x - 10 x - 5 \cdot - 5 - 0.5 x - 4 x^{2} < - 16$

Step 2.2.3.1.6

Multiply $- 5$ by $- 5$ .

$4 x^{2} - 10 x - 10 x + 25 - 0.5 x - 4 x^{2} < - 16$

Step 2.2.3.2

Subtract $10 x$ from $- 10 x$ .

$4 x^{2} - 20 x + 25 - 0.5 x - 4 x^{2} < - 16$

Step 2.3

Combine the opposite terms in $4 x^{2} - 20 x + 25 - 0.5 x - 4 x^{2}$ .

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Step 2.3.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$- 20 x + 25 - 0.5 x + 0 < - 16$

Step 2.3.2

Add $- 20 x + 25 - 0.5 x$ and $0$ .

$- 20 x + 25 - 0.5 x < - 16$

Step 2.4

Subtract $0.5 x$ from $- 20 x$ .

$- 20.5 x + 25 < - 16$

Step 3

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

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Step 3.1

Subtract $25$ from both sides of the inequality.

$- 20.5 x < - 16 - 25$

Step 3.2

Subtract $25$ from $- 16$ .

$- 20.5 x < - 41$

Step 4

Divide each term in $- 20.5 x < - 41$ by $- 20.5$ and simplify.

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Step 4.1

Divide each term in $- 20.5 x < - 41$ by $- 20.5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 20.5 x}{- 20.5} > \frac{- 41}{- 20.5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 20.5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 20.5 x}{- 20.5} > \frac{- 41}{- 20.5}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 41}{- 20.5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 41$ by $- 20.5$ .

$x > 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x > 2$

Interval Notation:

$(2, \infty)$

Step 6