4. Polynomial and Rational Functions

 Problems Modeled by Polynomial or Rational Inequalities

Polynomial and Rational Inequalities

Polynomial Applications (Maximizing Volume)

Answer the following. Why must -4 be in the solution set of x+4 / 2x+1 ≥ 0? (Do not solve the inequality.)

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 3x^2+16x<−5

Solving a Rational Inequality, More Examples - Example 2

Application with Polynomials

Solving Quadratic Inequalities, More Examples - Example 1

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2-7x+10>0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x+1)(x−7)≤0

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. x/(x + 2) ≥ 2

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x−4)(x+2)>0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3≥9x^2

Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. (x - 4)^2 ≤ 0

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (4−2x)/(3x+4)≤0

 Solving a Cubic Inequality Using a Sign Chart

Solve each quadratic inequality. Give the solution set in interval notation. See
Example 1. (a) -(x + 1)(x + 2) ≥ 0 (b) -(x + 1)(x + 2) > 0 (c) -(x + 1)(x + 2) ≤ 0 (d) -(x + 1)(x + 2) < 0

Solve each inequality in Exercises 86–91 using a graphing utility.

2x^2 + 5x - 3 ≤ 0

Solve each inequality in Exercises 65–70 and graph the solution set on a real number line.

3/(x +3) > 3/(x - 2)

Rational Function Application

Rational Function Application: Function Value, Equation, End Behavior

Solve each rational inequality. Give the solution set in interval notation.
(5x-3)^3/(25-8x)^2≤0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 6x^2+x>1

Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. x^2 - 2 > x

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. (x+2)/(2x+3)≤5

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2-x-6>0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. −x^2+x≥0

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (3x+5)/(6−2x)≥0

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5

Solve each inequality in Exercises 65–70 and graph the solution set on a real number line.

|x^2 + 2x - 36| > 12

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)(x−5)>0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. −x^2+2x≥0

In Exercises 73–74, use the graph of the rational function to solve each inequality. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/4b7667a3547e9973-1675173163525.jpg" alt="" />

1/4(x + 2) > - 3/4(x - 2)<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/003cd72c514dbb12-1678176664737.jpg" style="width: 300px;">

Solving a Rational Inequality

In Exercises 61–64, find the domain of each function.

f(x) = √(2x^2 - 5x + 2)

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3≤4x^2

Solve each rational inequality. Give the solution set in interval notation.
(9x-8)/(4x^2+25)<0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x+1)(x+2)(x+3)≥0

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. 4/(x+1)<2/(x+3)

Solving Rational Inequalities

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 4x^2+1≥4x

Solving Quadratic Inequalities - Example 1

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (1−x)^2(x−5/2)<0

Which inequality has solution set (-∞, ∞)? 
A. (x-3)^2≥0 B. (5x-6)^2≤0 C. (6x+4)^2>0 D. (8x+7)^2<0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^2+2x<0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 5x≤2−3x^2

Solve each inequality in Exercises 86–91 using a graphing utility.

(x - 4)/(x - 1) ≤ 0

Solving a Rational Inequality, More Examples - Example 3

Solving Quadratic Inequalities - A Common Mistake

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^2+5x+4>0

❖ Solving Linear Inequalities Made Easy ❖

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9.
5/(1-x)≤2/(1-x)

Solve each rational inequality. Give the solution set in interval notation. (9x-11)(2x+7)/(3x-8)^3>0

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9.
3/(2x-1)>-4/x

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2≤9

Solving Quadratic Inequalities, More Examples - Example 4

Solve each rational inequality. Give the solution set in interval notation.
(2x-3)(3x+8)/(x-6)^3≥0

Solving a Quadratic Inequality Using a Sign Chart

Solving a Quadratic Inequality Using Its Graph

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^2≤4x−2

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3+2x^2−4x−8≥0

In Exercises 71–72, use the graph of the polynomial function to solve each inequality. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/12615990e9a0f5a2-1674230236373.jpg" alt="" />

2x^3 + 11x^2 < 7x + 6<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/4696d17639d95464-1678176593567.jpg" style="width: 300px;">

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. 2x^2-9x≤18

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (−x+2)/(x−4)≥0

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+4)(x−1)/(x+2)≤0

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x(x-1)≤6

Solving Quadratic Inequalities - More Examples

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)/(x+4)<0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (2−x)^2(x−7/2)<0

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2+3x-4<0

Solving Quadratic Inequalities

Solve each inequality. Give the solution set using interval notation. x²-3x ≥ 5

Solving Quadratic Inequalities - Example 2

Rational Functions with Inequalities

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x(4−x)(x−6)≤0

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x(x+1)<12

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x−1)(x−2)(x−3)≥0

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. 2x^2 + 5x - 3 < 0

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. 4/(2-x)≥3/(1-x)

Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. 2^x2 + 5 ≤ 11x

Solving a Rational Inequality, More Examples - Example 1

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9.
1/(x+2)≥3

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 9x^2+3x−2≥0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3+x^2+4x+4>0

Solve each quadratic inequality. Give the solution set in interval notation. See
Example 1. (a) (x - 5)(x + 2) ≥ 0 (b) (x - 5)(x + 2) > 0 (c) (x - 5)(x + 2) ≤ 0 (d) (x - 5)(x + 2) < 0

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x - 2)/(x + 2) ≤ 2

Answer the following. Why can 3 not be in the solution set of 14x+9 / x-3 < 0? (Do not solve the inequality.)

In Exercises 73–74, use the graph of the rational function to solve each inequality. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/0add976dc0638875-1675172769959.jpg" alt="" />

1/4(x + 2) ≤ - 3/4(x - 2)<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/7ea802782057f468-1678176656340.jpg" style="width: 300px;">

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 2x^2+3x>0

Solving a Rational Inequality - Example 3

In Exercises 71–72, use the graph of the polynomial function to solve each inequality. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/e908c2b1aec32fb8-1674229730020.jpg" alt="" />

2x^3 + 11x^2 ≥ 7x + 6<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/4b1e9809049cef2c-1678176574990.jpg" style="width: 300px;">

Solve each rational inequality. Give the solution set in interval notation.
(5-3x)^2/(2x-5)^3>0

Solve each inequality. Give the solution set using interval notation. 3x+6 / x-5 > 0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3−3x^2−9x+27<0

Solve each rational inequality. Give the solution set in interval notation.
(2x-3)/x^2+1≥0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 4x^2−4x+1≥0

Solve each inequality in Exercises 86–91 using a graphing utility.

x^2 + 3x - 10 > 0

Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. x^2 + x - 30 ≤ 0

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2>16

Solving a Rational Inequality - Example 1

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (−x−3)/(x+2)≤0

Solve each inequality in Exercises 86–91 using a graphing utility.

1/(x + 1) ≤ 2/(x + 4)

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. 2x^2 + 9x + 4 ≥ 0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3+2x^2−x−2≥0

Solving and Graphing Polynomial and Rational Inequalities

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+4)/x>0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x(3−x)(x−5)≤0

Solve each inequality in Exercises 86–91 using a graphing utility.

x^3 + x^2 - 4x - 4 > 0

Solve each inequality. Give the solution set using interval notation. x+7 / 2x+1 ≤1

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^2≤2x+2

Solve each inequality. Give the solution set using interval notation. 3x-2/x - 4 > 0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 3x^2+10x−8≤0

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x−4)/(x+3) > 0

In Exercises 61–64, find the domain of each function.

f(x) = √(2x/(x + 1) - 1)

Solve each problem. See Example 7. Velocity of an Object The velocity of an object, v, after t seconds is given by v=3t^2-18t+24.Find the interval where the velocity is negative.

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^2−6x+9<0

Solving 3 Basic Linear Inequalities

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. 3x^2+x≤4

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 2x^2+x<15

Solving Quadratic Inequalities - Example 3

Solve each problem. See Example 7. A baseball is hit so that its height, s, in feet after t seconds is s=-16t^2+44t+4. For what time period is the ball at least 32 ft above the ground?

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 9x^2−6x+1<0

Solve each polynomial inequality. Give the solution set in interval notation. See
Examples 2 and 3. 23. (a) -x(x - 1)(x - 2) ≥ 0 (b) -x(x - 1)(x - 2) > 0 (c) -x(x - 1)(x - 2) ≤ 0 (d) -x(x - 1)(x - 2) < 0

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9.
4/(3+x)≤3/(3+x)

In Exercises 61–64, find the domain of each function.

f(x) = 1/√(4x^2 - 9x + 2)

Solve each inequality in Exercises 65–70 and graph the solution set on a real number line.

(x^2 -x - 2)/(x^2 - 4x + 3) > 0

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9.
5/(x+1)>12/(x+1)

In Exercises 61–64, find the domain of each function.

f(x) = √(x/(2x - 1) - 1)

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^2−4x≥0

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. (x+3)/(2x-5)≤1

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2-7x+10<0

Solving a Rational Inequality - Example 2

Solve each inequality in Exercises 65–70 and graph the solution set on a real number line.

1/(x + 1) > 2/(x - 1)

Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. -(x + 1)^2 ≥ 0

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 3x^2−5x≤0

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. x/(x−3)>0

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9.
7/(x+2)≥1/(x+2)

Solving Quadratic Inequalities, More Examples - Example 2

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