SOLVING RATIONAL INEQUALITIES WORKSHEET

Solve each inequality.

Problem 1 :

(x - 7)/(x - 1) < 0

Solution

Problem 2 :

(x + 5)/(x - 4) ≤ 0

Solution

Problem 3 :

(x + 32)/(x + 6) ≤ 3

Solution

Problem 4 :

(x + 68)/(x + 8) ≥ 5

Solution

Problem 5 :

(x + 3) (x + 5) / (x + 2) ≥ 0

Solution

Solve the following rational inequalities.

Example 1 :

(x + 6) / (x² - 5x - 24) ≥ 0

Solution

Example 2 :

-10/(x - 5) ≥ -11/(x - 6)

Solution

Example 3:

-3/(x + 7) ≤ -4/(x + 8)

Solution

Example 4 :

-7/(x + 5) ≤ -8/(x + 6)

Solution

Example 5 :

(x + 7) (x - 3) / (x - 5)² > 0

Solution

Solve for the following quadratic inequalities.

Problem 1 :

y² - y ≥ 12

Solution

Problem 2 :

-y² + 4 > 0

Solution

Problem 3 :

-y² + 3y - 2 ≤ 0

Solution

Problem 4 :

-2y² + 5y + 12 ≤ 0

Solution

Problem 5 :

Solve the inequality 3x² − 5x − 1 < 4x² + 7x + 19

Solution

Problem 6 :

Solve 2x² + 13x – 17 ≥ x² + 4x + 5.

Solution

Problem 1 :

Given 𝐴 = 𝑥² − 7 and 𝐵 = −4𝑥 + 5, for what values of 𝑥 is 𝐴 < 𝐵?

Solution

Problem 2 :

When a projectile is fired into the air, its height h, in meters, t seconds later is given by the equation ℎ(𝑡) = 11𝑡 − 3𝑡². When is the projectile at least 6 m above the ground?

Solution

Problem 3 :

The arch of the Sydney Harbor Bridge in Sydney, Australia, can be modeled by

y = −0.00211x² + 1.06x

where x is the distance (in meters) from the left pylons and y is the height (in meters) of the arch above the water. For what distances x is the arch above the road?

Solution

Problem 4 :

The number T of teams that have participated in a robot-building competition for high-school students over a recent period of time x (in years) can be modeled by

T(x) = 17.155 x² + 193.68 x + 235.81, 0 ≤ x ≤ 6.

After how many years is the number of teams greater than 1000? Justify your answer.

Solution

Example 5 :

A rectangular fountain display has a perimeter of 400 feet and an area of at least 9100 feet. Describe the possible widths of the fountain.

Solution

Solve for the following quadratic inequalities :

Problem 1 :

5 - 4y - y² > 0

Solution

Problem 2 :

1 - y - 2y² < 0

Solution

Problem 3 :

(y - 3) (y + 2) > 0

Solution

Problem 4 :

y² - 2y - 3 < 0

Solution

Problem 5 :

Find the set of values of x for which x² − 2x − 24 < 0 and 12 − 5x ≥ x + 9

Solution

Problem 6 :

Find the set of values of x for which x² − 100 > 0 and x² + 8x − 105 > 0

Solution

Make a sign chart to solve the inequalities. Give answers in interval solution.

Problem 1 :

x² + 9x - 22 < 0

Solution

Problem 2 :

x² - 16 ≥ 0

Solution

Problem 3 :

x(x - 3)² (x + 2) ≤ 0

Solution

Problem 4 :

(x - 1) / (x + 4) ≤ 0

Solution

Problem 5 :

x² / (x - 1) ≥ 0

Solution

Problem 6 :

(x² -4x + 3) / (x² + 4x - 21) > 0

Solution