7.6 Solve Rational Inequalities

Before you get started, take this readiness quiz.

Find the value of $x - 5$ when ⓐ $x = 6$ ⓑ $x = −3$ ⓒ $x = 5 .$
If you missed this problem, review the reference.

Solve: $8 - 2 x < 12 .$
If you missed this problem, review the reference.

Write in interval notation: $−3 \leq x < 5 .$
If you missed this problem, review the reference.

Solve and write the solution in interval notation: $\frac{x - 2}{x + 4} \geq 0 .$

Solve and write the solution in interval notation: $\frac{x + 2}{x - 4} \geq 0 .$

Solve and write the solution in interval notation: $\frac{3 x}{x - 3} < 1 .$

Solve and write the solution in interval notation: $\frac{3 x}{x - 4} < 2 .$

Solve and write the solution in interval notation: $\frac{1}{x^{2} + 2 x - 8} > 0 .$

Solve and write the solution in interval notation: $\frac{3}{x^{2} + x - 12} > 0 .$

Solve and write the solution in interval notation: $\frac{1}{2} + \frac{4}{x^{2}} < \frac{3}{x} .$

Solve and write the solution in interval notation: $\frac{1}{3} + \frac{6}{x^{2}} < \frac{3}{x} .$

Given the function $R \left(x\right) = \frac{x - 2}{x + 4} ,$ find the values of x that make the function less than or equal to 0.

Given the function $R \left(x\right) = \frac{x + 1}{x - 4} ,$ find the values of x that make the function less than or equal to 0.

The function $C \left(x\right) = 10 x + 3000$ represents the cost to produce $x ,$ number of items. Find ⓐ the average cost function, $c \left(x\right)$ ⓑ how many items should be produced so that the average cost is less than $40.

The function $C \left(x\right) = 20 x + 6000$ represents the cost to produce $x ,$ number of items. Find ⓐ the average cost function, $c \left(x\right)$ ⓑ how many items should be produced so that the average cost is less than $60?

The function $C \left(x\right) = 5 x + 900$ represents the cost to produce $x ,$ number of items. Find ⓐ the average cost function, $c \left(x\right)$ ⓑ how many items should be produced so that the average cost is less than $20?

Solve each rational inequality and write the solution in interval notation.

$\frac{x - 3}{x + 4} \geq 0$

Solve each rational inequality and write the solution in interval notation.

$\frac{x + 1}{x - 3} \leq 0$

Solve each rational inequality and write the solution in interval notation.

$\frac{x - 7}{x - 1} > 0$

Solve each rational inequality and write the solution in interval notation.

$\frac{x - 6}{x + 5} < 0$

Solve each rational inequality and write the solution in interval notation.

$\frac{3 x}{x - 5} < 1$

Solve each rational inequality and write the solution in interval notation.

$\frac{6 x}{x - 6} > 2$

Solve each rational inequality and write the solution in interval notation.

$\frac{2 x + 3}{x - 6} \leq 1$

Solve each rational inequality and write the solution in interval notation.

$\frac{3 x - 2}{x - 4} \geq 2$

Solve each rational inequality and write the solution in interval notation.

$\frac{1}{x^{2} + 7 x + 12} > 0$

Solve each rational inequality and write the solution in interval notation.

$\frac{3}{x^{2} - 5 x + 4} < 0$

Solve each rational inequality and write the solution in interval notation.

$\frac{2}{2 x^{2} + x - 15} \geq 0$

Solve each rational inequality and write the solution in interval notation.

$\frac{−2}{6 x^{2} - 13 x + 6} \leq 0$

Solve each rational inequality and write the solution in interval notation.

$\frac{1}{2} + \frac{12}{x^{2}} > \frac{5}{x}$

Solve each rational inequality and write the solution in interval notation.

$\frac{1}{2} - \frac{4}{x^{2}} \leq \frac{1}{x}$

Solve each rational inequality and write the solution in interval notation.

$\frac{1}{x^{2} - 16} < 0$

Solve each rational inequality and write the solution in interval notation.

$\frac{4}{x - 2} \geq \frac{3}{x + 1}$

Solve each rational function inequality and write the solution in interval notation.

Given the function $R \left(x\right) = \frac{x - 5}{x - 2} ,$ find the values of $x$ that make the function less than or equal to 0.

Solve each rational function inequality and write the solution in interval notation.

Given the function $R \left(x\right) = \frac{x - 6}{x + 2}$, find the values of x that make the function less than or equal to 0.

Write the steps you would use to explain solving rational inequalities to your little brother.

Determine the values for which the rational expression is undefined.

$\frac{5 a + 3}{3 a - 2}$

Determine the values for which the rational expression is undefined.

$\frac{5 x^{2} y^{2}}{8 y}$

Simplify.

$\frac{18}{24}$

Simplify.

$\frac{x^{2} + 7 x + 12}{x^{2} + 8 x + 16}$

Multiply.

$\frac{5}{8} \cdot \frac{4}{15}$

Multiply.

$\frac{72 x - 12 x^{2}}{8 x + 32} \cdot \frac{x^{2} + 10 x + 24}{x^{2} - 36}$

Divide.

$\frac{x^{2} - 4 x - 12}{x^{2} + 8 x + 12} \div \frac{x^{2} - 36}{3 x}$

Divide.

$\frac{11 + w}{w - 9} \div \frac{121 - w^{2}}{9 - w}$

Divide.

$\frac{\frac{c^{2} - 64}{3 c^{2} + 26 c + 16}}{\frac{c^{2} - 4 c - 32}{15 c + 10}}$

Multiply and Divide Rational Functions

Find $R \left(x\right) = f \left(x\right) \cdot g \left(x\right)$ where $f \left(x\right) = \frac{9 x^{2} + 9 x}{x^{2} - 3 x - 4}$ and $g \left(x\right) = \frac{x^{2} - 16}{3 x^{2} + 12 x} .$

Perform the indicated operations.

$\frac{7}{15} + \frac{8}{15}$

Perform the indicated operations.

$\frac{y^{2} + 10 y}{y + 5} + \frac{25}{y + 5}$

Perform the indicated operations.

$\frac{x^{2}}{x - 7} - \frac{3 x + 28}{x - 7}$

Perform the indicated operations.

$\frac{4 q^{2} - q + 3}{q^{2} + 6 q + 5} - \frac{3 q^{2} + q + 6}{q^{2} + 6 q + 5}$

Add and subtract.

$\frac{18 w}{6 w - 1} + \frac{3 w - 2}{1 - 6 w}$

Add and subtract.

$\frac{2 b^{2} + 3 b - 15}{b^{2} - 49} - \frac{b^{2} + 16 b - 1}{49 - b^{2}}$

Find the LCD.

$\frac{7}{a^{2} - 3 a - 10} , \frac{3 a}{a^{2} - a - 20}$

Find the LCD.

$\frac{5}{3 p^{2} + 17 p - 6} , \frac{2 m}{3 p^{2} + 23 p - 8}$

Perform the indicated operations.

$\frac{2}{c - 2} + \frac{9}{c + 3}$

Perform the indicated operations.

$\frac{2 x}{x^{2} + 10 x + 24} + \frac{3 x}{x^{2} + 8 x + 16}$

Perform the indicated operations.

$\frac{3 y}{y + 2} - \frac{y + 2}{y + 8}$

Perform the indicated operations.

$\frac{7 m + 3}{m + 2} - 5$

Perform the indicated operations.

$\frac{8 a}{a^{2} - 64} - \frac{4}{a + 8}$

Find $R \left(x\right) = f \left(x\right) + g \left(x\right)$ where $f \left(x\right)$ and $g \left(x\right)$ are given.

$f \left(x\right) = \frac{2 x^{2} + 12 x - 11}{x^{2} + 3 x - 10} ,$$g \left(x\right) = \frac{x + 1}{2 - x}$

Find $R \left(x\right) = f \left(x\right) - g \left(x\right)$ where $f \left(x\right)$ and $g \left(x\right)$ are given.

$f \left(x\right) = \frac{4 x}{x^{2} - 121} ,$$g \left(x\right) = \frac{2}{x - 11}$

Simplify.

$\frac{\frac{7 x}{x + 2}}{\frac{14 x^{2}}{x^{2} - 4}}$

Simplify.

$\frac{x - \frac{3 x}{x + 5}}{\frac{1}{x + 5} + \frac{1}{x - 5}}$

Simplify.

$\frac{\frac{1}{3} + \frac{1}{8}}{\frac{1}{4} + \frac{1}{12}}$

Simplify.

$\frac{\frac{2}{z^{2} - 49} + \frac{1}{z + 7}}{\frac{9}{z + 7} + \frac{12}{z - 7}}$

Solve.

$\frac{1}{2} + \frac{2}{3} = \frac{1}{x}$

Solve.

$\frac{1}{b - 2} + \frac{1}{b + 2} = \frac{3}{b^{2} - 4}$

Solve.

$\frac{v - 15}{v^{2} - 9 v + 18} = \frac{4}{v - 3} + \frac{2}{v - 6}$

Solve Rational Equations that Involve Functions

For rational function, $f \left(x\right) = \frac{x + 2}{x^{2} - 6 x + 8} ,$ ⓐ find the domain of the function ⓑ solve $f \left(x\right) = 1$ ⓒ find the points on the graph at this function value.

Solve for the indicated variable.

$\frac{V}{l} = h w$ for $l .$

Solve for the indicated variable.

$x = \frac{y + 5}{z - 7}$ for $z .$

Solve.

$\frac{x}{4} = \frac{3}{5}$

Solve.

$\frac{s}{s + 20} = \frac{3}{7}$

Solve.

Rachael had a 21-ounce strawberry shake that has 739 calories. How many calories are there in a 32-ounce shake?

Solve.

Francesca is 5.75 feet tall. Late one afternoon, her shadow was 8 feet long. At the same time, the shadow of a nearby tree was 32 feet long. Find the height of the tree.

Solve.

When making the 5-hour drive home from visiting her parents, Lolo ran into bad weather. She was able to drive 176 miles while the weather was good, but then driving 10 mph slower, went 81 miles when it turned bad. How fast did she drive when the weather was bad?

Solve.

Josue can ride his bicycle 8 mph faster than Arjun can ride his bike. It takes Arjun 3 hours longer than Josue to ride 48 miles. How fast can Josue ride his bike?

Solve.

Brandy can frame a room in 1 hour, while Jake takes 4 hours. How long could they frame a room working together?

Solve.

Jeffrey can paint a house in 6 days, but if he gets a helper he can do it in 4 days. How long would it take the helper to paint the house alone?

Solve.

If $y$ varies directly as $x$ when $y = 9$ and $x = 3 ,$ find $x$ when $y = 21 .$

Solve.

Vanessa is traveling to see her fiancé. The distance, $d ,$ varies directly with the speed, $v ,$ she drives. If she travels 258 miles driving 60 mph, how far would she travel going 70 mph?

Solve.

The distance to stop a car varies directly with the square of its speed. It takes 200 feet to stop a car going 50 mph. How many feet would it take to stop a car going 60 mph?

Solve.

The number of tickets for a music fundraiser varies inversely with the price of the tickets. If Madelyn has just enough money to purchase 12 tickets for $6 each, how many tickets can Madelyn afford to buy if the price increased to $8?

Solve each rational inequality and write the solution in interval notation.

$\frac{x - 3}{x + 4} \leq 0$

Solve each rational inequality and write the solution in interval notation.

$\frac{3 x - 2}{x - 4} \leq 2$

Solve each rational inequality and write the solution in interval notation.

$\frac{1}{2} - \frac{4}{x^{2}} \geq \frac{1}{x}$

Solve each rational function inequality and write the solution in interval notation

Given the function, $R \left(x\right) = \frac{x - 5}{x - 2} ,$ find the values of $x$ that make the function greater than or equal to 0.

Solve each rational function inequality and write the solution in interval notation

The function
$C \left(x\right) = 150 x + 100,000$ represents the cost to produce $x ,$ number of items. Find ⓐ the average cost function, $c \left(x\right)$ ⓑ how many items should be produced so that the average cost is less than $160.

Simplify.

$\frac{4 a^{2} b}{12 a b^{2}}$

Perform the indicated operation and simplify.

$\frac{4 x}{x + 2} \cdot \frac{x^{2} + 5 x + 6}{12 x^{2}}$

Perform the indicated operation and simplify.

$\frac{6 x^{2} - x + 20}{x^{2} - 81} - \frac{5 x^{2} + 11 x - 7}{x^{2} - 81}$

Perform the indicated operation and simplify.

$\frac{2 n^{2} + 8 n - 1}{n^{2} - 1} - \frac{n^{2} - 7 n - 1}{1 - n^{2}}$

Perform the indicated operation and simplify.

$\frac{\frac{1}{m} - \frac{1}{n}}{\frac{1}{n} + \frac{1}{m}}$

Solve each equation.

$\frac{1}{z - 5} + \frac{1}{z + 5} = \frac{1}{z^{2} - 25}$

Solve each rational inequality and write the solution in interval notation.

$\frac{6 x}{x - 6} \leq 2$

Solve each rational inequality and write the solution in interval notation.

$\frac{1}{2} + \frac{12}{x^{2}} \geq \frac{5}{x}$

Find $R \left(x\right)$ given $f \left(x\right) = \frac{x - 4}{x^{2} - 3 x - 10}$ and $g \left(x\right) = \frac{x - 5}{x^{2} - 2 x - 8} .$

$R \left(x\right) = f \left(x\right) \cdot g \left(x\right)$

Find $R \left(x\right)$ given $f \left(x\right) = \frac{x - 4}{x^{2} - 3 x - 10}$ and $g \left(x\right) = \frac{x - 5}{x^{2} - 2 x - 8} .$

Given the function,
$R \left(x\right) = \frac{2}{2 x^{2} + x - 15} ,$ find the values of $x$ that make the function less than or equal to 0.

Solve.

If $y$ varies inversely with the square of $x$ and $x = 3$ when $y = 9 ,$ find $y$ when $x = 4 .$

Solve.

Oliver can split a truckload of logs in 8 hours, but working with his dad they can get it done in 3 hours. How long would it take Oliver’s dad working alone to split the logs?