A type of inequality involving at least one rational expression
Example 1: Find the solution for the inequality 2x/(x + 1) < 1
The answer in a rational inequality is called the solution set, which can be an interval of values
How to solve rational inequalities
1. Rewrite the inequality as a single fraction on one side and zero on the other
2. Determine critical values where the rational expression is zero or undefined
3. Mark critical values on a number line to partition into intervals
4. Select a test point in each interval to determine positive and negative values
5. Determine the solution set based on the sign of the rational expression
Intervals and Descriptions
(-∞, 2) x < 2
(1, +∞) x > 1
(-5,2) -5 < x < 2
(-∞, -4] x ≤ -4
[0, +∞) x ≥ 0
(-3,1] -3 < x ≤ 1
Rational Inequalities
x > 9
x/2x
x + 1 < 1
x - 2 ≥ 1
x/(x^2 + 3x + 2)
x^2 - 16 ≤ 0
Learning Objectives: At the end of this lesson, you should be able to solve rational inequalities
Topics to revisit before starting the lesson
Fractions and Rational Expressions
Operations on Fractions
Finding the Least Common Denominator
Interval Notation
Solving Inequalities
Solving Rational Inequalities
Step 1: Rewrite as a single fraction being compared to zero. Rewrite the inequality where all the terms are on one side leaving zero on the other side. Combine the terms as one rational expression using the least common denominator (LCD). The resulting inequality will be the basis for the succeeding steps.
Solving Rational Inequalities
Step 2: Determine the critical values. Determining the critical values means finding the zeroes of the numerator and denominator. The zeroes of a polynomial are the values of the unknown making the value of the polynomial equal to zero. Thus, finding the zeroes of a polynomial is by equating the polynomial to zero then solve for the unknown.
Solving Rational Inequalities
Step 3: List the intervals. The number line will be partitioned by the critical values (CV), thus we will have the following intervals: -∞, -1; -1, 1; 1, +∞
Solving Rational Inequalities
Example 1: Find the solution for the inequality 2𝑥 / (𝑥 + 1) < 1
Solving Rational Inequalities
Step 4: Select a test point and create a table of signs. A test point (TP) is any number within an interval. These test points will be substituted in the numerator and in the denominator, and the resulting sign shall be noted. The signs at these test points are also the signs in their aforementioned intervals.
Solving Rational Inequalities
1. Select a test point and create a table of signs
2. Substitute the test points and critical values to the unknown in the numerator and denominator, and take note of the resulting sign
3. Divide the resulting signs to determine the sign of the rational expression in that interval
4. Determine the solution set
Test Points and Critical Values
-2
-1
0
1
2
Intervals and Critical Values
-∞, -1
-1
-1,1
1
1, +∞
When the critical values are substituted into the rational expression, the zero of the numerator, 1, makes the rational expression equal to 0, while the zero of the denominator, -1, makes the rational expression undefined
The inequality 𝑥 − 1 / 𝑥 + 1 < 0 means that the expression 𝑥−1 / 𝑥+1 must have a value less than 0, meaning negative
The number line will be partitioned by the critical values, yielding the following intervals
The expression 𝑥−1/𝑥+1 must have a value less than 0, meaning negative
The solution set for the given inequality 2𝑥/𝑥+1 < 1 is -1, 1
The resulting 𝑥^2−9/𝑥 > 0 will be the basis in finding the solution set of the given rational inequality
Based on the table of signs, the rational expression has a positive value in the intervals -3,0 and 3, +∞. And, it has a negative value in the intervals -∞, -3 and 0,3
Find the solution for the inequality 𝑥 > 9/𝑥
The critical values in the rational expression are -3, 3, and 0
Rationality in classical economic theory is a flawed assumption as people usually don't act rationally
When analysing markets, a range of assumptions are made about the rationality of economic agents involved in the transactions
Since 𝑥2−9𝑥 > 0 in the intervals −3,0 and 3, +∞ or for when −3 < 𝑥 < 0 or �� > 3, therefore, the solution set for the given inequality 𝑥 > 9𝑥 is −3, 0 ∪ (3, +∞)
The Wealth of Nations was written
1776
Example 3: Find the solution for the inequality 𝑥2 + 3𝑥 + 2𝑥2 − 16 ≤ 0
Moreover, when we substitute the critical values on the rational expression, the zeroes of the numerator, −3 and 3, make the rational expression equal to 0, while the zero of the denominator, 0, makes rational expression undefined
Notice the use of the symbol ∪ (union) to combine two separate intervals as one solution set
Solving Rational Inequalities
1. Step 4: Select a test point and create a table of signs
2. Step 5: Determine the solution set
Rational agents will select the choice which presents the highest benefits
Solving Rational Inequalities
1. Step 1: Rewrite as a single fraction being compared to zero
2. Step 2: Determine the critical values
The rational expression has a positive value in the intervals −3,0 and 3, +∞. And, it has a negative value in the intervals −∞, −3 and 0,3
Solving Rational Inequalities
Step 2: Determine the critical values
Based on the table of signs, the rational expression has a positive value in the intervals -∞, -4, -2, -1, and 4, +∞. And, it has a negative value in the intervals -4, -2, and -1, 4
Solving Rational Inequalities
Step 3: List the intervals
Moreover, when we substitute the critical values on the rational expression, the zeroes of the numerator, -2 and -1, make the rational expression equal to 0, while the zeroes of the denominator, -4 and 4, make the rational expression undefined