Algebra Examples

$\frac{5}{4} - \frac{4}{3 y} \leq \frac{4}{2}$

Step 1

Divide $4$ by $2$ .

$\frac{5}{4} - \frac{4}{3 y} \leq 2$

Step 2

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

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Step 2.1

Subtract $\frac{5}{4}$ from both sides of the inequality.

$- \frac{4}{3 y} \leq 2 - \frac{5}{4}$

Step 2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{4}{3 y} \leq 2 \cdot \frac{4}{4} - \frac{5}{4}$

Step 2.3

Combine $2$ and $\frac{4}{4}$ .

$- \frac{4}{3 y} \leq \frac{2 \cdot 4}{4} - \frac{5}{4}$

Step 2.4

Combine the numerators over the common denominator.

$- \frac{4}{3 y} \leq \frac{2 \cdot 4 - 5}{4}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $2$ by $4$ .

$- \frac{4}{3 y} \leq \frac{8 - 5}{4}$

Step 2.5.2

Subtract $5$ from $8$ .

$- \frac{4}{3 y} \leq \frac{3}{4}$

Step 3

Multiply both sides by $3 y$ .

$- \frac{4}{3 y} (3 y) = \frac{3}{4} (3 y)$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $3 y$ .

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Step 4.1.1.1

Move the leading negative in $- \frac{4}{3 y}$ into the numerator.

$\frac{- 4}{3 y} (3 y) = \frac{3}{4} (3 y)$

Step 4.1.1.2

Cancel the common factor.

$\frac{- 4}{3 y} (3 y) = \frac{3}{4} (3 y)$

Step 4.1.1.3

Rewrite the expression.

$- 4 = \frac{3}{4} (3 y)$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $\frac{3}{4} (3 y)$ .

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Step 4.2.1.1

Combine $3$ and $\frac{3}{4}$ .

$- 4 = \frac{3 \cdot 3}{4} y$

Step 4.2.1.2

Multiply $3$ by $3$ .

$- 4 = \frac{9}{4} y$

Step 4.2.1.3

Combine $\frac{9}{4}$ and $y$ .

$- 4 = \frac{9 y}{4}$

Step 5

Solve for $y$ .

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Step 5.1

Rewrite the equation as $\frac{9 y}{4} = - 4$ .

$\frac{9 y}{4} = - 4$

Step 5.2

Multiply both sides of the equation by $\frac{4}{9}$ .

$\frac{4}{9} \cdot \frac{9 y}{4} = \frac{4}{9} \cdot - 4$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{4}{9} \cdot \frac{9 y}{4}$ .

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Step 5.3.1.1.1

Cancel the common factor of $4$ .

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Step 5.3.1.1.1.1

Cancel the common factor.

$\frac{4}{9} \cdot \frac{9 y}{4} = \frac{4}{9} \cdot - 4$

Step 5.3.1.1.1.2

Rewrite the expression.

$\frac{1}{9} (9 y) = \frac{4}{9} \cdot - 4$

Step 5.3.1.1.2

Cancel the common factor of $9$ .

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Step 5.3.1.1.2.1

Factor $9$ out of $9 y$ .

$\frac{1}{9} (9 (y)) = \frac{4}{9} \cdot - 4$

Step 5.3.1.1.2.2

Cancel the common factor.

$\frac{1}{9} (9 y) = \frac{4}{9} \cdot - 4$

Step 5.3.1.1.2.3

Rewrite the expression.

$y = \frac{4}{9} \cdot - 4$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $\frac{4}{9} \cdot - 4$ .

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Step 5.3.2.1.1

Multiply $\frac{4}{9} \cdot - 4$ .

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Step 5.3.2.1.1.1

Combine $\frac{4}{9}$ and $- 4$ .

$y = \frac{4 \cdot - 4}{9}$

Step 5.3.2.1.1.2

Multiply $4$ by $- 4$ .

$y = \frac{- 16}{9}$

Step 5.3.2.1.2

Move the negative in front of the fraction.

$y = - \frac{16}{9}$

Step 6

Find the domain of $- \frac{4}{3 y} - \frac{3}{4}$ .

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Step 6.1

Set the denominator in $\frac{4}{3 y}$ equal to $0$ to find where the expression is undefined.

$3 y = 0$

Step 6.2

Divide each term in $3 y = 0$ by $3$ and simplify.

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Step 6.2.1

Divide each term in $3 y = 0$ by $3$ .

$\frac{3 y}{3} = \frac{0}{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{0}{3}$

Step 6.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{3}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $0$ by $3$ .

$y = 0$

Step 6.3

The domain is all values of $y$ that make the expression defined.

$(- \infty, 0) \cup (0, \infty)$

Step 7

Use each root to create test intervals.

$y < - \frac{16}{9}$

$- \frac{16}{9} < y < 0$

$y > 0$

Step 8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 8.1

Test a value on the interval $y < - \frac{16}{9}$ to see if it makes the inequality true.

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Step 8.1.1

Choose a value on the interval $y < - \frac{16}{9}$ and see if this value makes the original inequality true.

$y = - 4$

Step 8.1.2

Replace $y$ with $- 4$ in the original inequality.

$\frac{5}{4} - \frac{4}{3 (- 4)} \leq \frac{4}{2}$

Step 8.1.3

The left side $1.58\overline{3}$ is less than the right side $2$ , which means that the given statement is always true.

True

Step 8.2

Test a value on the interval $- \frac{16}{9} < y < 0$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $- \frac{16}{9} < y < 0$ and see if this value makes the original inequality true.

$y = - 1$

Step 8.2.2

Replace $y$ with $- 1$ in the original inequality.

$\frac{5}{4} - \frac{4}{3 (- 1)} \leq \frac{4}{2}$

Step 8.2.3

The left side $2.58\overline{3}$ is greater than the right side $2$ , which means that the given statement is false.

False

Step 8.3

Test a value on the interval $y > 0$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $y > 0$ and see if this value makes the original inequality true.

$y = 2$

Step 8.3.2

Replace $y$ with $2$ in the original inequality.

$\frac{5}{4} - \frac{4}{3 (2)} \leq \frac{4}{2}$

Step 8.3.3

The left side $0.58\overline{3}$ is less than the right side $2$ , which means that the given statement is always true.

True

Step 8.4

Compare the intervals to determine which ones satisfy the original inequality.

$y < - \frac{16}{9}$ True

$- \frac{16}{9} < y < 0$ False

$y > 0$ True

$y < - \frac{16}{9}$ True

$- \frac{16}{9} < y < 0$ False

$y > 0$ True

Step 9

The solution consists of all of the true intervals.

$y \leq - \frac{16}{9}$ or $y > 0$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$y \leq - \frac{16}{9} or y > 0$

Interval Notation:

$(- \infty, - \frac{16}{9}] \cup (0, \infty)$

Step 11