Linear Inequalities and Absolute Value Inequalities
It is not easy to make the honor role at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.![Several red winner’s ribbons lie on a white table.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/09/25200407/CNX_CAT_Figure_02_07_001N.jpg)
Write and Manipulate Inequalities
Indicating the solution to an inequality such as can be achieved in several ways. We can use a number line as shown below. The blue ray begins at and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.![A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/24225859/CNX_CAT_Figure_02_07_002.jpg)
Set Indicated | Set-Builder Notation | Interval Notation |
---|---|---|
All real numbers between a and b, but not including a or b | ||
All real numbers greater than a, but not including a | ||
All real numbers less than b, but not including b | ||
All real numbers greater than a, including a | ||
All real numbers less than b, including b | ||
All real numbers between a and b, including a | ||
All real numbers between a and b, including b | ||
All real numbers between a and b, including a and b | ||
All real numbers less than a or greater than b | ||
All real numbers |
Example: Using Interval Notation to Express All Real Numbers Greater Than or Equal to a
Use interval notation to indicate all real numbers greater than or equal to .Answer: Use a bracket on the left of and parentheses after infinity: . The bracket indicates that is included in the set with all real numbers greater than to infinity.
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Use interval notation to indicate all real numbers between and including and .Answer:
Example: Using Interval Notation to Express All Real Numbers Less Than or Equal to a or Greater Than or Equal to b
Write the interval expressing all real numbers less than or equal to or greater than or equal to .Answer: We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at and ends at , which is written as . The second interval must show all real numbers greater than or equal to , which is written as . However, we want to combine these two sets. We accomplish this by inserting the union symbol, , between the two intervals.
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Express all real numbers less than or greater than or equal to 3 in interval notation.Answer:
Example: Demonstrating the Multiplication Property
Illustrate the multiplication property for inequalities by solving each of the following:Answer:
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Solve .Answer:
Example: Solving an Inequality with Fractions
Solve the following inequality and write the answer in interval notation: .Answer: We begin solving in the same way we do when solving an equation.
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Solve the inequality and write the answer in interval notation: .Answer:
Example: Solving a Compound Inequality with the Variable in All Three Parts
Solve the compound inequality with variables in all three parts: .Answer: Lets try the first method. Write two inequalities:
![A number line with the points -4 and 5/6 labeled. Dots appear at these points and a line connects these two dots.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/24225901/CNX_CAT_Figure_02_07_003.jpg)
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Solve the compound inequality: .Answer:
Key Concepts
- Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well.
- Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality.
- Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities.
- Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value.
- Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution.