Algebra Examples

$\frac{x}{a^{2} - b^{2}} + \frac{3 x}{a - b} < \frac{5}{a + b}$

Step 1

Subtract $\frac{5}{a + b}$ from both sides of the inequality.

$\frac{x}{a^{2} - b^{2}} + \frac{3 x}{a - b} - \frac{5}{a + b} < 0$

Step 2

Simplify $\frac{x}{a^{2} - b^{2}} + \frac{3 x}{a - b} - \frac{5}{a + b}$ .

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Step 2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = b$ .

$\frac{x}{(a + b) (a - b)} + \frac{3 x}{a - b} - \frac{5}{a + b} < 0$

Step 2.2

To write $\frac{3 x}{a - b}$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

$\frac{x}{(a + b) (a - b)} + \frac{3 x}{a - b} \cdot \frac{a + b}{a + b} - \frac{5}{a + b} < 0$

Step 2.3

Write each expression with a common denominator of $(a + b) (a - b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{3 x}{a - b}$ by $\frac{a + b}{a + b}$ .

$\frac{x}{(a + b) (a - b)} + \frac{3 x (a + b)}{(a - b) (a + b)} - \frac{5}{a + b} < 0$

Step 2.3.2

Reorder the factors of $(a - b) (a + b)$ .

$\frac{x}{(a + b) (a - b)} + \frac{3 x (a + b)}{(a + b) (a - b)} - \frac{5}{a + b} < 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x + 3 x (a + b)}{(a + b) (a - b)} - \frac{5}{a + b} < 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $x$ out of $x + 3 x (a + b)$ .

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Step 2.5.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} + 3 x (a + b)}{(a + b) (a - b)} - \frac{5}{a + b} < 0$

Step 2.5.1.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 + 3 x (a + b)}{(a + b) (a - b)} - \frac{5}{a + b} < 0$

Step 2.5.1.3

Factor $x$ out of $3 x (a + b)$ .

$\frac{x \cdot 1 + x (3 (a + b))}{(a + b) (a - b)} - \frac{5}{a + b} < 0$

Step 2.5.1.4

Factor $x$ out of $x \cdot 1 + x (3 (a + b))$ .

$\frac{x (1 + 3 (a + b))}{(a + b) (a - b)} - \frac{5}{a + b} < 0$

Step 2.5.2

Apply the distributive property.

$\frac{x (1 + 3 a + 3 b)}{(a + b) (a - b)} - \frac{5}{a + b} < 0$

Step 2.6

To write $- \frac{5}{a + b}$ as a fraction with a common denominator, multiply by $\frac{a - b}{a - b}$ .

$\frac{x (1 + 3 a + 3 b)}{(a + b) (a - b)} - \frac{5}{a + b} \cdot \frac{a - b}{a - b} < 0$

Step 2.7

Multiply $\frac{5}{a + b}$ by $\frac{a - b}{a - b}$ .

$\frac{x (1 + 3 a + 3 b)}{(a + b) (a - b)} - \frac{5 (a - b)}{(a + b) (a - b)} < 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{x (1 + 3 a + 3 b) - 5 (a - b)}{(a + b) (a - b)} < 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{x \cdot 1 + x (3 a) + x (3 b) - 5 (a - b)}{(a + b) (a - b)} < 0$

Step 2.9.2

Simplify.

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Step 2.9.2.1

Multiply $x$ by $1$ .

$\frac{x + x (3 a) + x (3 b) - 5 (a - b)}{(a + b) (a - b)} < 0$

Step 2.9.2.2

Rewrite using the commutative property of multiplication.

$\frac{x + 3 x a + x (3 b) - 5 (a - b)}{(a + b) (a - b)} < 0$

Step 2.9.2.3

Rewrite using the commutative property of multiplication.

$\frac{x + 3 x a + 3 x b - 5 (a - b)}{(a + b) (a - b)} < 0$

Step 2.9.3

Apply the distributive property.

$\frac{x + 3 x a + 3 x b - 5 a - 5 (- b)}{(a + b) (a - b)} < 0$

Step 2.9.4

Multiply $- 1$ by $- 5$ .

$\frac{x + 3 x a + 3 x b - 5 a + 5 b}{(a + b) (a - b)} < 0$

Step 3

Multiply both sides by $(a + b) (a - b)$ .

$\frac{x + 3 x a + 3 x b - 5 a + 5 b}{(a + b) (a - b)} ((a + b) (a - b)) < 0 ((a + b) (a - b))$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{x + 3 x a + 3 x b - 5 a + 5 b}{(a + b) (a - b)} ((a + b) (a - b))$ .

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Step 4.1.1.1

Cancel the common factor of $(a + b) (a - b)$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{x + 3 x a + 3 x b - 5 a + 5 b}{(a + b) (a - b)} ((a + b) (a - b)) < 0 ((a + b) (a - b))$

Step 4.1.1.1.2

Rewrite the expression.

$x + 3 x a + 3 x b - 5 a + 5 b < 0 ((a + b) (a - b))$

Step 4.1.1.2

Reorder.

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Step 4.1.1.2.1

Move $x$ .

$x + 3 a x + 3 x b - 5 a + 5 b < 0 ((a + b) (a - b))$

Step 4.1.1.2.2

Move $x$ .

$x + 3 a x + 3 b x - 5 a + 5 b < 0 ((a + b) (a - b))$

Step 4.1.1.2.3

Move $x$ .

$3 a x + 3 b x - 5 a + 5 b + x < 0 ((a + b) (a - b))$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $0 ((a + b) (a - b))$ .

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Step 4.2.1.1

Expand $(a + b) (a - b)$ using the FOIL Method.

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Step 4.2.1.1.1

Apply the distributive property.

$3 a x + 3 b x - 5 a + 5 b + x < 0 (a (a - b) + b (a - b))$

Step 4.2.1.1.2

Apply the distributive property.

$3 a x + 3 b x - 5 a + 5 b + x < 0 (a \cdot a + a (- b) + b (a - b))$

Step 4.2.1.1.3

Apply the distributive property.

$3 a x + 3 b x - 5 a + 5 b + x < 0 (a \cdot a + a (- b) + b a + b (- b))$

Step 4.2.1.2

Simplify terms.

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Step 4.2.1.2.1

Combine the opposite terms in $a \cdot a + a (- b) + b a + b (- b)$ .

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Step 4.2.1.2.1.1

Reorder the factors in the terms $a (- b)$ and $b a$ .

$3 a x + 3 b x - 5 a + 5 b + x < 0 (a \cdot a - a b + a b + b (- b))$

Step 4.2.1.2.1.2

Add $- a b$ and $a b$ .

$3 a x + 3 b x - 5 a + 5 b + x < 0 (a \cdot a + 0 + b (- b))$

Step 4.2.1.2.1.3

Add $a \cdot a$ and $0$ .

$3 a x + 3 b x - 5 a + 5 b + x < 0 (a \cdot a + b (- b))$

Step 4.2.1.2.2

Simplify each term.

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Step 4.2.1.2.2.1

Multiply $a$ by $a$ .

$3 a x + 3 b x - 5 a + 5 b + x < 0 (a^{2} + b (- b))$

Step 4.2.1.2.2.2

Rewrite using the commutative property of multiplication.

$3 a x + 3 b x - 5 a + 5 b + x < 0 (a^{2} - b \cdot b)$

Step 4.2.1.2.2.3

Multiply $b$ by $b$ by adding the exponents.

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Step 4.2.1.2.2.3.1

Move $b$ .

$3 a x + 3 b x - 5 a + 5 b + x < 0 (a^{2} - (b \cdot b))$

Step 4.2.1.2.2.3.2

Multiply $b$ by $b$ .

$3 a x + 3 b x - 5 a + 5 b + x < 0 (a^{2} - b^{2})$

Step 4.2.1.2.3

Multiply $0$ by $a^{2} - b^{2}$ .

$3 a x + 3 b x - 5 a + 5 b + x < 0$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Add $5 a$ to both sides of the inequality.

$3 a x + 3 b x + 5 b + x < 5 a$

Step 5.1.2

Subtract $5 b$ from both sides of the inequality.

$3 a x + 3 b x + x < 5 a - 5 b$

Step 5.2

Factor $x$ out of $3 a x + 3 b x + x$ .

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Step 5.2.1

Factor $x$ out of $3 a x$ .

$x (3 a) + 3 b x + x < 5 a - 5 b$

Step 5.2.2

Factor $x$ out of $3 b x$ .

$x (3 a) + x (3 b) + x < 5 a - 5 b$

Step 5.2.3

Raise $x$ to the power of $1$ .

$x (3 a) + x (3 b) + x < 5 a - 5 b$

Step 5.2.4

Factor $x$ out of $x^{1}$ .

$x (3 a) + x (3 b) + x \cdot 1 < 5 a - 5 b$

Step 5.2.5

Factor $x$ out of $x (3 a) + x (3 b)$ .

$x (3 a + 3 b) + x \cdot 1 < 5 a - 5 b$

Step 5.2.6

Factor $x$ out of $x (3 a + 3 b) + x \cdot 1$ .

$x (3 a + 3 b + 1) < 5 a - 5 b$

Step 5.3

Divide each term in $x (3 a + 3 b + 1) < 5 a - 5 b$ by $3 a + 3 b + 1$ and simplify.

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Step 5.3.1

Divide each term in $x (3 a + 3 b + 1) < 5 a - 5 b$ by $3 a + 3 b + 1$ .

$\frac{x (3 a + 3 b + 1)}{3 a + 3 b + 1} < \frac{5 a}{3 a + 3 b + 1} + \frac{- 5 b}{3 a + 3 b + 1}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $3 a + 3 b + 1$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{x (3 a + 3 b + 1)}{3 a + 3 b + 1} < \frac{5 a}{3 a + 3 b + 1} + \frac{- 5 b}{3 a + 3 b + 1}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x < \frac{5 a}{3 a + 3 b + 1} + \frac{- 5 b}{3 a + 3 b + 1}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Combine the numerators over the common denominator.

$x < \frac{5 a - 5 b}{3 a + 3 b + 1}$

Step 5.3.3.2

Factor $5$ out of $5 a - 5 b$ .

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Step 5.3.3.2.1

Factor $5$ out of $5 a$ .

$x < \frac{5 (a) - 5 b}{3 a + 3 b + 1}$

Step 5.3.3.2.2

Factor $5$ out of $- 5 b$ .

$x < \frac{5 (a) + 5 (- b)}{3 a + 3 b + 1}$

Step 5.3.3.2.3

Factor $5$ out of $5 (a) + 5 (- b)$ .

$x < \frac{5 (a - b)}{3 a + 3 b + 1}$