1. Solving Nonlinear Inequalities
Digital Lesson
Solving Nonlinear Inequalities

2. The symbols , , and  may also be used.
A quadratic inequality in one variable is an inequality which can be written in the form
ax2 + bx + c > 0 (a  0)
for a, b, c real numbers.
The symbols , , and  may also be used.
Example: x2 – 3x + 7  0 is a quadratic inequality since it can be written 1x2 + (– 3)x + 7  0.
Example: 3x2 < x + 5 is a quadratic inequality since it can be written 3x2 + (–1)x + (– 5) < 0.
Example: x2 + 3x  x is not a quadratic inequality since it is equivalent to 3x  4  0.
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Quadratic Inequality

3. Example: Which of the values of x are solutions of x2 + 3x  4  0 ?
A solution of a quadratic inequality in one variable is a number which, when substituted for the variable, results in a true inequality.
Example: Which of the values of x are solutions of x2 + 3x  4  0 ?
x x2 + 3x – x2 + 3x – 4  Solution?
 1
(1)2 + 3(1) – 4
 6  0
true
yes
(0)2 + 3(0) – 4
 4  0
true
yes
0.5
(0.5)2 + 3(0.5) – 4
 2.25  0
true
yes 1
(1)2 + 3(1) – 4
0  0
true
yes 2
(2)2 + 3(2) – 4
6  0
false no 3
(3)2 + 3(3) – 4
14  0
false no
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Solution

4. [ ] The solution set of an inequality is the set of all solutions.
Study the graph of the solution set of x2 + 3x  4  0. [ ] -2 -1 1 2 -6 -5
- 4 -3
The solution set is {x |  4  x  1}.
The values of x for which equality holds are part of the solution set.
These values can be found by solving the quadratic equation associated with the inequality.
x2 + 3x  4 = 0
Solve the associated equation.
(x + 4)(x  1) = 0
Factor the trinomial.
x =  4 or x = 1
Solutions of the equation
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Solution Set

5. Solving a Quadratic Inequality
To solve a quadratic inequality:
1. If necessary, rewrite the quadratic inequality so that zero appears on the right, then factor.
2. On the real number line, draw a vertical line at the numbers that make each factor equal to zero.
3. For each factor, place plus signs above the number line in the regions where the factor is positive, and minus signs where the factor is negative.
4. Observe the sign of the product of the factors for each region, to determine which regions will belong to the solution set.
5. Express the solution set using set-builder notation and a graph on a real number line.
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Solving a Quadratic Inequality

6. Example: Solve and Graph the Solution Set
Example: Solve and graph the solution set of x2  6x + 5 < 0.
The product of the factors is negative.
(x  1)(x  5) < 0
Factor.
x  1 = 0
x  5 = 0
Solve for each factor equal to zero.
x = 1
x = 5
Draw vertical lines indicating the numbers where each factor equals zero.
Product is positive.
Product is negative.
Product is positive.
x – 1
x – 5
– – –
+ + +
For each region, identify if each factor is positive or negative.
– – –
– – – – –
+ + + ( )
Factors 3 4 5 6 7 -1 1 2
Draw the solution set. Rounded parentheses indicate a strict inequality.
{x | 1 < x < 5}
Solution set in set-builder notation.
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Example: Solve and Graph the Solution Set

7. Example: Solve and Graph the Solution Set
Example: Solve and graph the solution set of x2  x  6.
x2  x  6  0
Rewrite the inequality so that zero appears on the right.
The product of the factors is positive.
(x + 2)(x  3)  0
Factor.
Numbers where each factor equals zero.
x =  2, 3
x + 2
x – 3
– – – –
+ +
Draw vertical lines where each factor equals zero.
– – – –
– – – – – – –
+ + ] [
Indicate positive and negative regions for each factor. 1 2 3 4
- 4 -3 -2 -1
Square brackets are used since the inequality is .
Draw solution set.
{x | x   2 or x  3}
Solution set in set-builder notation.
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Example: Solve and Graph the Solution Set

8. Example: Solve a Cubic Inequality
Cubic inequalities can be solved similarly.
Example: Solve and graph the solution set of x3 + x2  9x  9 > 0.
x2(x + 1)  9(x + 1) > 0
Factor by grouping.
(x2  9)(x + 1) > 0
(x + 3)(x  3)(x + 1) > 0
x =  3, +3, 1
Numbers where each factor equals zero.
x + 3
x + 1
x – 3
– –
– – – – –
+ +
Draw three vertical lines.
+ +
– –
Indicate positive and negative regions for each of the three factors.
– –
+ + ( ) ( 1 2 3 4
- 4 -3 -2 -1
{x | 3 < x < 1 or x > 3}
Solution set
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Example: Solve a Cubic Inequality

9. Example: Solve an Inequality Involving a Rational Function
Inequalities involving rational functions can be solved similarly.
Example: Solve and graph the solution set of
(x + 1) = 0
(x  2) = 0
Find the numbers for which each factor equals zero.
x =  1
x = 2
Note that 2 will not be part of the solution set since the expression is not defined when the denominator is zero.
x + 1
x – 2
– – – – –
+ + +
There are two regions where the quotient of the two factors is positive.
– – – –
– – – – –
+ + + ] ( 1 2 3 4
- 4 -3 -2 -1
{x | x   1 or x > 2}
Solution set
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Example: Solve an Inequality Involving a Rational Function

10. Example: Solve and Graph the Solution Set
Example: Solve and graph the solution set of
The quotient is negative.
Factor.
x + 2 = 0
(x  1)(x + 3) = 0
Expression is undefined at these points.
x = 2
x = 1, 3
x  1
x + 3
x + 2
– – – –
– –
– – – –
– – + ) ( ) 1 2 3 4
- 4 -3 -2 -1
{x | x < 3 or 2 < x < 1}
Solution set
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Example: Solve and Graph the Solution Set

11. Example: One leg of a right triangle is 2 inches longer than the other
Example: One leg of a right triangle is 2 inches longer than the other. How long should the shorter leg be to ensure that the area of the triangle is greater than or equal to 4?
x = shorter leg
 4
x + 2 = other leg x
x + 2
Area of triangle
Solve:
x + 4
x – 2 –
+ + +
– – – – – – – – –
+ + + ] [ 1 2 3 4
- 4 -3 -2 -1
Since length has to be positive, the answer is x  2.
The shorter leg should be at least 2 inches long.
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Example: Word Problem