Precalculus by Richard Wright

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The King will reply, ‘Truly I tell you, whatever you did for one of the least of these brothers and sisters of mine, you did for me.’ Matthew‬ ‭25‬:‭40‬ ‭NIV‬‬‬‬‬‬‬‬‬

2-09 Nonlinear Inequalities

Mr. Wright teaches the lesson.

Summary: In this section, you will:

SDA NAD Content Standards (2018): PC.6.2

Chicken farm
Figure 1: credit (Pixabay: tlemens)

The average cost of a dozen eggs from a small farm can be modeled by \(C(x) = \frac{0.364x+5790}{x}\), where x is the number of eggs produced in dozens. The farmer would like to keep the costs less than $0.75 per dozen eggs. How many eggs would his chickens have to produce?

This function is not linear and the solution will be an interval of answers since there is a range for the desired costs. The words "less than" indicate an inequality. Thus the function is an example of a nonlinear inequality.

Nonlinear Inequality

A nonlinear inequality is a mathematical statement containing an inequality such as < or > and the expression does not describe a straight line.

Find Critical Numbers

The first step to solve a nonlinear inequality is to find the critical numbers. These are input values that make the expression equal zero or be undefined.

Find Critical Numbers
  1. Make one side of the inequality equal zero.
  2. Find the zeros by replacing the inequality with an equal sign and solve for x.
  3. Find any x-values that cause the expression to be undefined, such as dividing by zero or square roots of negative numbers.

Example 1: Find Critical Numbers

The average cost of a dozen eggs from a small farm can be modeled by \(C(x) = \frac{0.364x+5790}{x}\), where x is the number of eggs produced in dozens. The farmer would like to keep the costs less than $0.75 per dozen eggs. How many eggs would his chickens have to produce?

Solution

Write the inequality based on the problem. The costs are to be less than $0.75, so the 0.75 is the biggest the costs can be. So, the big side of the inequality will face the 0.75.

$$ 0.75 > \frac{0.364x+5790}{x} $$

Make one side zero and simplify.

$$ 0 > \frac{0.364x+5790}{x} - 0.75 $$

$$ 0 > \frac{0.364x+5790}{x} - \frac{0.75x}{x} $$

$$ 0 > \frac{-0.386x+5790}{x} $$

Find the zeros by replacing the inequality with an equal sign and solve for x.

$$ 0 = \frac{-0.386x+5790}{x} $$

0 = -0.386x + 5790

−5790 = −0.386x

x = 15,000

Find any x-values that cause the expression to be undefined.

$$ 0 > \frac{-0.386x+5790}{x} $$

The function will be undefined if the denominator is equal to zero. So, x = 0 is a critical number.

The critical numbers are 0 and 15,000.

Try It 1

Find the critical numbers of 0 < 2x2 − 9x − 5.

Answer

−1⁄2, 5

Solve Nonlinear Inequalities Algebraically

Now that we have found the critical numbers, we can finish solving for x. The critical numbers are places where the function can change from the positive numbers to negative. In other words, the critical numbers show where the function changes between greater than zero and less than zero. If we test the intervals between the critical numbers we can see which are greater than zero and which are less than zero.

Solve Nonlinear Inequalities Algebraically
  1. Find the critical numbers.
  2. Graph the critical numbers on a number line.
  3. Choose a number to test in each interval between the critical numbers.
  4. Test these numbers in the inequality to see which produce true statements.
  5. The intervals that produce true statements are the solutions.

Example 2: Solve a Nonlinear Inequality

The average cost of a dozen eggs from a small farm can be modeled by \(C(x) = \frac{0.364x+5790}{x}\), where x is the number of eggs produced in dozens. The farmer would like to keep the costs less than $0.75 per dozen eggs. How many eggs would his chickens have to produce?

Solution

This was already rewritten this in example 1 as

$$ 0 > \frac{-0.386x+5790}{x} $$

and the critical numbers are 0 and 15,000.

Plot those critical numbers on a number line.

0 and 15000 on number line.
Figure 2

Choose a number in each interval between the critical numbers. −100, 5000, and 15100 will work.

Critical numbers 0 and 15000 on number line with the test values of −100, 5000, and 15100.
Figure 3

Test these in the inequality to see which produce true statements.

$$ -100: 0 > \frac{-0.386(-100)+5790}{-100} \rightarrow 0 > -58.286 \text{ True} $$

$$ 5000: 0 > \frac{-0.386(5000)+5790}{5000} \rightarrow 0 > 0.772 \text{ False} $$

$$ 15100: 0 > \frac{-0.386(15100)+5790}{15100} \rightarrow 0 > -0.00256 \text{ True} $$

The solutions are the true intervals: (−∞, 0) and (15,000, ∞). Since the hens cannot produce a negative number of eggs, the farm needs to produce more than 15,000 dozen eggs.

Analysis

Both ends of the intervals are parentheses because the inequality is not an equals inequality. If the inequality was ≤ or ≥, there would be a bracket at the end of the interval. However, critical numbers that cause the expression to be undefined always are given a parenthesis in interval notation because x cannot be those numbers.

Example 3: Solve a Nonlinear Inequality Algebraically

Solve \(-2 ≤ \frac{2x^2-12}{x}\).

Solution

Solve one side for zero.

$$ 0 ≤ \frac{2x^2-12}{x} + 2 $$

$$ 0 ≤ \frac{2x^2-12}{x} + \frac{2x}{x} $$

$$ 0 ≤ \frac{2x^2+2x-12}{x} $$

Find the critical numbers.

$$ 0 = \frac{2x^2+2x-12}{x} $$

0 = 2x2 + 2x − 12

0 = 2(x2 + x − 6)

0 = 2(x + 3)(x − 2)

x = −3, 2

0 is also a critical number because the inequality is undefined when x = 0.

Graph the critical numbers on a number line and pick test numbers in each interval. −4, −1, 1 and 3 will work for test numbers.

Critical numbers and test numbers on a number line.
Figure 4

Test the test numbers in the solved inequality.

$$ -4: 0 ≤ \frac{2(-4)^2+2(-4)-12}{(-4)} \rightarrow 0 ≤ -3 \text{ False} $$

$$ -1: 0 ≤ \frac{2(-1)^2+2(-1)-12}{(-1)} \rightarrow 0 ≤ 12 \text{ True} $$

$$ 1: 0 ≤ \frac{2(1)^2+2(1)-12}{(1)} \rightarrow 0 ≤ -8 \text{ False} $$

$$ 3: 0 ≤ \frac{2(3)^2+2(3)-12}{(3)} \rightarrow 0 ≤ 4 \text{ True} $$

The solutions are [−3, 0) and [2, ∞).

Analysis

The critical number 0 comes from when the inequality is undefined, so x ≠ 0. That is why the 0 has a parenthesis by it. The −3 and 2 have brackets by them because x can equal them from the ≤ sign.

Try It 2

Solve 0 < 2x2 − 9x − 5.

Answer

(−1⁄2, 5)

Solve Nonlinear Inequalities by Graphing

An alternate way to solve nonlinear inequalities is by graphing. Remember a coordinate plane is simply two number lines set perpendicularly. By solving the inequality so that one side is zero and graphing the expression, the function can be compared with zero. The critical numbers are the zeros and the undefined values. The zeros occur at the x-intercepts; the undefined values often occur at vertical asymptotes. These can both be read from the graph. Intervals on the graph where the y-values are positive are where the inequality is greater than zero. Negative y-values show the intervals where the inequality is less than zero.

Solve Nonlinear Inequalities by Graphing
  1. Solve the inequality so one side is zero.
  2. Graph the resulting expression from the nonzero side of the inequality.
  3. If the inequality is > 0, the solutions are intervals with positive y-values.
  4. If the inequality is < 0, the solutions are intervals with negative y-values.

Example 4: Solve Nonlinear Inequality by Graphing

Solve 8 ≥ x2 − 2x.

Solution

Solve the inequality so one side is zero.

8 ≥ x2 − 2x
0 ≥ x2 − 2x − 8

Graph the right-hand expression.

y=x^2-2x-8
Figure 5

Since the expression is less than zero, the solutions are intervals with negative y-values. The solution is [−2, 4].

Example 5: Solve Nonlinear Inequality by Graphing

Solve \(0 < \frac{x-3}{x^2+x-2}\).

Solution

The inequality is already solved for zero, so graph the right-hand expression.

y=(x-3)/(x^2+x-2)
Figure 6

Since the expression is greater than zero, the solutions are intervals with positive y-values. The solutions are (−2, 1) ∪ (3, ∞).

Try It 3

Solve \(0 > \frac{x^2}{x-2}\) by graphing.

Answer

(2, ∞)

Lesson Summary

Nonlinear Inequality

A nonlinear inequality is a mathematical statement containing an inequality such as < or > and the expression does not describe a straight line.


Find Critical Numbers
  1. Make one side of the inequality equal zero.
  2. Find the zeros by replacing the inequality with an equal sign and solve for x.
  3. Find any x-values that cause the expression to be undefined, such as dividing by zero or square roots of negative numbers.

Solve Nonlinear Inequalities Algebraically
  1. Find the critical numbers.
  2. Graph the critical numbers on a number line.
  3. Choose a number to test in each interval between the critical numbers.
  4. Test these numbers in the inequality to see which produce true statements.
  5. The intervals that produce true statements are the solutions.

Solve Nonlinear Inequalities by Graphing
  1. Solve the inequality so one side is zero.
  2. Graph the resulting expression from the nonzero side of the inequality.
  3. If the inequality is > 0, the solutions are intervals with positive y-values.
  4. If the inequality is < 0, the solutions are intervals with negative y-values.

Helpful videos about this lesson.

Practice Exercises

  1. When do you use brackets instead of parentheses when writing intervals?
  2. Find the critical numbers.

  3. f(x) = x2 + 5x − 24
  4. g(x) = x3 + 3x2 − 4x
  5. \(h(x) = \frac{x^2+4x+4}{x-1}\)
  6. Solve inequalities algebraically.

  7. x3 − 4x2 − 11x < −30
  8. 2 ≥ 6x2x
  9. \(0 ≤ \frac{2x}{2x^2+x-1}\)
  10. \(1 > \frac{3}{x+1}\)
  11. \(0 ≤ \frac{x^2+2x-35}{x-2}\)
  12. Solve by graphing.

  13. 0 ≥ x2 + 6x + 9
  14. 0 < x3 − 7x + 6
  15. \(0 ≤ \frac{x}{x^2-3x+2}\)
  16. \(0 > \frac{x^2+4x+3}{x-3}\)
  17. \(0 < \frac{x+1}{x^3+2x^2-8x}\)
  18. Problem Solving

  19. It costs a computer software company $13,000 to code a new piece of office software and $2.00 to print and package each disc.
    1. Write a function for the average cost of each disk.
    2. How many discs do they have to sell to make the average cost less than $15?
  20. Mixed Review

  21. (2-08) Graph \(f(x) = \frac{1}{x^2+4}\).
  22. (2-06) Find the zeros of 2x3 + 5x2x − 6.
  23. (2-04) Divide (2x3x + 10) ÷ (x2 + 2x − 1).
  24. (2-02) Find the vertex of y = 2x2 − 4x + 3.
  25. (2-01) Multiply (3 + i)(1 − 3i).

Answers

  1. Brackets are when the end of the interval is included such as ≤ or ≥. Parentheses are when the end of the interval is not included such as < or >.
  2. −8, 3
  3. −4, 0, 1
  4. −2, 1
  5. (−∞, −3) ∪ (2, 5)
  6. [−½, ⅔]
  7. (−1, 0] ∪ (½, ∞)
  8. (−∞, −1) ∪ (2, ∞)
  9. [−7, 2) ∪ [5, ∞)
  10. −3
  11. (−3, 1) ∪ (2, ∞)
  12. [0, 1) ∪ (2, ∞)
  13. (−∞, −3) ∪ (−1, 3)
  14. (−∞, −4) ∪ (−1, 0) ∪ (2, ∞)
  15. a. \(C(x) = \frac{2x+13000}{x}\); b. More than 1000 discs
  16. −2, \(-\frac{3}{2}\), 1
  17. \(2x - 4 + \frac{9x+6}{x^2+2x-1}\)
  18. (1, 1)
  19. 6 − 8i