Rational Inequalities 5/4-4/(3y)

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

Divide $4$ by $2$ .

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

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

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

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

Simplify the right side of the inequality.

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To write $\frac{2}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{4}{3 y} \leq \frac{2}{1} \cdot \frac{4}{4} - \frac{5}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$- \frac{4}{3 y} \leq \frac{2 \cdot 4}{1 \cdot 4} - \frac{5}{4}$

Multiply $4$ by $1$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $2$ by $4$ .

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

Subtract $5$ from $8$ .

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

Solve for $y$ .

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Multiply each term by $y$ and simplify.

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Multiply each term in $- \frac{4}{3 y} \leq \frac{3}{4}$ by $y$ .

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

Simplify the left side of the inequality by cancelling the common factors.

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Cancel the common factor of $y$ .

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

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

Factor $y$ out of $3 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Move $\frac{3 y}{4}$ to the left side of the equation by subtracting it from both sides.

$- \frac{4}{3} - \frac{3 y}{4} \leq 0$

Find the LCD of the terms in the equation.

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To find the LCD of a set of numbers $(- \frac{4}{3}, - \frac{3 y}{4})$ , find the LCM of the denominators.

$LCM (3, 4)$

Calculate the LCM of first two denominators in the list, $3$ and $4$ .

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Find the values of the numerical part of each term. Select the largest one, which in this case is $4$ . Multiply them together to get the current total. In this case, the current total is $12$ .

Current Total = $12$

Multiply the numeric part of the denominators together.

Current Total = $4 + 4 = 8$

Multiply the numeric part of the denominators together.

Current Total = $8 + 4 = 12$

Check each value in the numerical part of each term against the current total. Since the current total is evenly divisible, return it. That is the least common denominator of the numerical part of the fraction.

$12$

Multiply each term by $12$ and simplify.

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Multiply each term in $- \frac{4}{3} - \frac{3 y}{4} \leq 0$ by $12$ .

$- \frac{4}{3} \cdot 12 - \frac{3 y}{4} \cdot 12 \leq 0 \cdot 12$

Simplify each term.

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Cancel the common factor of $3$ .

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

$\frac{- 4}{3} \cdot 12 - \frac{3 y}{4} \cdot 12 \leq 0 \cdot 12$

Factor $3$ out of $12$ .

$\frac{- 4}{3} \cdot (3 (4)) - \frac{3 y}{4} \cdot 12 \leq 0 \cdot 12$

Cancel the common factor.

$\frac{- 4}{3} \cdot (3 \cdot 4) - \frac{3 y}{4} \cdot 12 \leq 0 \cdot 12$

Rewrite the expression.

$- 4 \cdot 4 - \frac{3 y}{4} \cdot 12 \leq 0 \cdot 12$

Multiply $- 4$ by $4$ .

$- 16 - \frac{3 y}{4} \cdot 12 \leq 0 \cdot 12$

Cancel the common factor of $4$ .

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

$- 16 + \frac{- 3 y}{4} \cdot 12 \leq 0 \cdot 12$

Factor $4$ out of $12$ .

$- 16 + \frac{- 3 y}{4} \cdot (4 (3)) \leq 0 \cdot 12$

Cancel the common factor.

$- 16 + \frac{- 3 y}{4} \cdot (4 \cdot 3) \leq 0 \cdot 12$

Rewrite the expression.

$- 16 - 3 y \cdot 3 \leq 0 \cdot 12$

Multiply $3$ by $- 3$ .

$- 16 - 9 y \leq 0 \cdot 12$

Multiply $0$ by $12$ .

$- 16 - 9 y \leq 0$

Add $16$ to both sides of the inequality.

$- 9 y \leq 16$

Divide each term by $- 9$ and simplify.

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Divide each term in $- 9 y \leq 16$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Cancel the common factor of $- 9$ .

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Cancel the common factor.

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

Divide $y$ by $1$ .

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

Move the negative in front of the fraction.

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

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

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

$3 y = 0$

Divide each term by $3$ and simplify.

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

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

Cancel the common factor of $3$ .

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Cancel the common factor.

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

Divide $y$ by $1$ .

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

Divide $0$ by $3$ .

$y = 0$

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

Interval Notation:

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

Interval Notation:

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

Use each root to create test intervals.

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

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

$y > 0$

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

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

$y = - 4$

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

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

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

True

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

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

$y = - 1$

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

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

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

False

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

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

$y = 2$

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

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

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

True

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

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

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

$y > 0$ True

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

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

$y > 0$ True

The solution consists of all of the true intervals.

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

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Rational Inequalities 5/4-4/(3y)<=4/2

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