Pre-Algebra Examples

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

Step 1

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

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Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

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

Step 1.3

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

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 1.4.1.2

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

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Step 1.4.1.2.1

Move $x$ .

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

Step 1.4.1.2.2

Multiply $x$ by $x$ .

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

Step 1.4.1.3

Multiply $2$ by $4$ .

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

Step 1.4.1.4

Multiply $5$ by $2$ .

$8 x^{2} + 10 x - 3 (4 x) - 3 \cdot 5 < (8 x + 1) (x - 7)$

Step 1.4.1.5

Multiply $4$ by $- 3$ .

$8 x^{2} + 10 x - 12 x - 3 \cdot 5 < (8 x + 1) (x - 7)$

Step 1.4.1.6

Multiply $- 3$ by $5$ .

$8 x^{2} + 10 x - 12 x - 15 < (8 x + 1) (x - 7)$

Step 1.4.2

Subtract $12 x$ from $10 x$ .

$8 x^{2} - 2 x - 15 < (8 x + 1) (x - 7)$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Apply the distributive property.

$8 x^{2} - 2 x - 15 < 8 x (x - 7) + 1 (x - 7)$

Step 2.1.2

Apply the distributive property.

$8 x^{2} - 2 x - 15 < 8 x \cdot x + 8 x \cdot - 7 + 1 (x - 7)$

Step 2.1.3

Apply the distributive property.

$8 x^{2} - 2 x - 15 < 8 x \cdot x + 8 x \cdot - 7 + 1 x + 1 \cdot - 7$

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 $x$ by adding the exponents.

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Step 2.2.1.1.1

Move $x$ .

$8 x^{2} - 2 x - 15 < 8 (x \cdot x) + 8 x \cdot - 7 + 1 x + 1 \cdot - 7$

Step 2.2.1.1.2

Multiply $x$ by $x$ .

$8 x^{2} - 2 x - 15 < 8 x^{2} + 8 x \cdot - 7 + 1 x + 1 \cdot - 7$

Step 2.2.1.2

Multiply $- 7$ by $8$ .

$8 x^{2} - 2 x - 15 < 8 x^{2} - 56 x + 1 x + 1 \cdot - 7$

Step 2.2.1.3

Multiply $x$ by $1$ .

$8 x^{2} - 2 x - 15 < 8 x^{2} - 56 x + x + 1 \cdot - 7$

Step 2.2.1.4

Multiply $- 7$ by $1$ .

$8 x^{2} - 2 x - 15 < 8 x^{2} - 56 x + x - 7$

Step 2.2.2

Add $- 56 x$ and $x$ .

$8 x^{2} - 2 x - 15 < 8 x^{2} - 55 x - 7$

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.

$8 x^{2} - 2 x - 15 - 8 x^{2} < - 55 x - 7$

Step 3.2

Add $55 x$ to both sides of the inequality.

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

Step 3.3

Combine the opposite terms in $8 x^{2} - 2 x - 15 - 8 x^{2} + 55 x$ .

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Step 3.3.1

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

$- 2 x - 15 + 0 + 55 x < - 7$

Step 3.3.2

Add $- 2 x - 15$ and $0$ .

$- 2 x - 15 + 55 x < - 7$

Step 3.4

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

$53 x - 15 < - 7$

Step 4

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

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Step 4.1

Add $15$ to both sides of the inequality.

$53 x < - 7 + 15$

Step 4.2

Add $- 7$ and $15$ .

$53 x < 8$

Step 5

Divide each term in $53 x < 8$ by $53$ and simplify.

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Step 5.1

Divide each term in $53 x < 8$ by $53$ .

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

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.

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

Step 5.2.1.2

Divide $x$ by $1$ .

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

Step 6

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Step 7