Algebra Examples

$(8 x + 1) (x - 7) \geq (2 x - 3) (4 x + 5)$

Step 1

Simplify $(8 x + 1) (x - 7)$ .

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Step 1.1

Rewrite.

$0 + 0 + (8 x + 1) (x - 7) \geq (2 x - 3) (4 x + 5)$

Step 1.2

Simplify by adding zeros.

$(8 x + 1) (x - 7) \geq (2 x - 3) (4 x + 5)$

Step 1.3

Expand $(8 x + 1) (x - 7)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$8 x (x - 7) + 1 (x - 7) \geq (2 x - 3) (4 x + 5)$

Step 1.3.2

Apply the distributive property.

$8 x \cdot x + 8 x \cdot - 7 + 1 (x - 7) \geq (2 x - 3) (4 x + 5)$

Step 1.3.3

Apply the distributive property.

$8 x \cdot x + 8 x \cdot - 7 + 1 x + 1 \cdot - 7 \geq (2 x - 3) (4 x + 5)$

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 $x$ by adding the exponents.

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Step 1.4.1.1.1

Move $x$ .

$8 (x \cdot x) + 8 x \cdot - 7 + 1 x + 1 \cdot - 7 \geq (2 x - 3) (4 x + 5)$

Step 1.4.1.1.2

Multiply $x$ by $x$ .

$8 x^{2} + 8 x \cdot - 7 + 1 x + 1 \cdot - 7 \geq (2 x - 3) (4 x + 5)$

Step 1.4.1.2

Multiply $- 7$ by $8$ .

$8 x^{2} - 56 x + 1 x + 1 \cdot - 7 \geq (2 x - 3) (4 x + 5)$

Step 1.4.1.3

Multiply $x$ by $1$ .

$8 x^{2} - 56 x + x + 1 \cdot - 7 \geq (2 x - 3) (4 x + 5)$

Step 1.4.1.4

Multiply $- 7$ by $1$ .

$8 x^{2} - 56 x + x - 7 \geq (2 x - 3) (4 x + 5)$

Step 1.4.2

Add $- 56 x$ and $x$ .

$8 x^{2} - 55 x - 7 \geq (2 x - 3) (4 x + 5)$

Step 2

Simplify $(2 x - 3) (4 x + 5)$ .

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Step 2.1

Expand $(2 x - 3) (4 x + 5)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$8 x^{2} - 55 x - 7 \geq 2 x (4 x + 5) - 3 (4 x + 5)$

Step 2.1.2

Apply the distributive property.

$8 x^{2} - 55 x - 7 \geq 2 x (4 x) + 2 x \cdot 5 - 3 (4 x + 5)$

Step 2.1.3

Apply the distributive property.

$8 x^{2} - 55 x - 7 \geq 2 x (4 x) + 2 x \cdot 5 - 3 (4 x) - 3 \cdot 5$

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

Rewrite using the commutative property of multiplication.

$8 x^{2} - 55 x - 7 \geq 2 \cdot 4 x \cdot x + 2 x \cdot 5 - 3 (4 x) - 3 \cdot 5$

Step 2.2.1.2

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

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Step 2.2.1.2.1

Move $x$ .

$8 x^{2} - 55 x - 7 \geq 2 \cdot 4 (x \cdot x) + 2 x \cdot 5 - 3 (4 x) - 3 \cdot 5$

Step 2.2.1.2.2

Multiply $x$ by $x$ .

$8 x^{2} - 55 x - 7 \geq 2 \cdot 4 x^{2} + 2 x \cdot 5 - 3 (4 x) - 3 \cdot 5$

Step 2.2.1.3

Multiply $2$ by $4$ .

$8 x^{2} - 55 x - 7 \geq 8 x^{2} + 2 x \cdot 5 - 3 (4 x) - 3 \cdot 5$

Step 2.2.1.4

Multiply $5$ by $2$ .

$8 x^{2} - 55 x - 7 \geq 8 x^{2} + 10 x - 3 (4 x) - 3 \cdot 5$

Step 2.2.1.5

Multiply $4$ by $- 3$ .

$8 x^{2} - 55 x - 7 \geq 8 x^{2} + 10 x - 12 x - 3 \cdot 5$

Step 2.2.1.6

Multiply $- 3$ by $5$ .

$8 x^{2} - 55 x - 7 \geq 8 x^{2} + 10 x - 12 x - 15$

Step 2.2.2

Subtract $12 x$ from $10 x$ .

$8 x^{2} - 55 x - 7 \geq 8 x^{2} - 2 x - 15$

Step 3

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

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Step 3.1

Subtract $8 x^{2}$ from both sides of the inequality.

$8 x^{2} - 55 x - 7 - 8 x^{2} \geq - 2 x - 15$

Step 3.2

Add $2 x$ to both sides of the inequality.

$8 x^{2} - 55 x - 7 - 8 x^{2} + 2 x \geq - 15$

Step 3.3

Combine the opposite terms in $8 x^{2} - 55 x - 7 - 8 x^{2} + 2 x$ .

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Step 3.3.1

Subtract $8 x^{2}$ from $8 x^{2}$ .

$- 55 x - 7 + 0 + 2 x \geq - 15$

Step 3.3.2

Add $- 55 x - 7$ and $0$ .

$- 55 x - 7 + 2 x \geq - 15$

Step 3.4

Add $- 55 x$ and $2 x$ .

$- 53 x - 7 \geq - 15$

Step 4

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

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Step 4.1

Add $7$ to both sides of the inequality.

$- 53 x \geq - 15 + 7$

Step 4.2

Add $- 15$ and $7$ .

$- 53 x \geq - 8$

Step 5

Divide each term in $- 53 x \geq - 8$ by $- 53$ and simplify.

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Step 5.1

Divide each term in $- 53 x \geq - 8$ by $- 53$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 53 x}{- 53} \leq \frac{- 8}{- 53}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 53$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 53 x}{- 53} \leq \frac{- 8}{- 53}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 8}{- 53}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{8}{53}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{8}{53}$

Interval Notation:

$(- \infty, \frac{8}{53}]$

Step 7