2.14: Algebraic Solutions to Absolute Value Inequalities
A ball is fired from the cannon during the Independence Day celebrations. It is fired directly into the air with an initial velocity of 150 ft/sec. The speed of the cannon ball can be calculated using the formula , where is the speed measure in ft/sec and t is the time in seconds. Calculate the times when the speed is greater than 86 ft/sec.
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Khan Academy Absolute Value Inequalities
Guidance
You have learned that a linear inequality is of the form , or . Linear inequalities, unlike linear equations, have more than one solution. They have a solution set. For example, if you look at the linear inequality . You know that is equal to 5, therefore the solution set could be any number greater than 2.
Recall that when solving absolute value linear equations, you have to solve for the two related equations. Remember that for , you had to solve for and . The same is true for linear inequalities. If you have an absolute value linear inequality, you would need to solve for the two related linear inequalities.
The table below shows the four types of absolute value linear inequalities and the two related inequality expressions required to be solved for each one.
Absolute Value Inequality | ||||
---|---|---|---|---|
Equation 1 | ||||
Equation 2 |
Remember the rules to algebraically solve for the variable remain the same as you have used before. It is also important to remember, as you found with absolute value linear equations, that if , there is no solution.
Example A
Solve for the absolute value inequality .
Solution:
Solutions and .
Example B
Solve for the absolute value inequality .
Solution:
Solutions and .
Example C
Solve for the absolute value inequality .
Solution:
Solutions and .
Concept Problem Revisited
A ball is fired from the cannon during the Independence Day celebrations. It is fired directly into the air with an initial velocity of 150 ft/sec. The speed of the cannon ball can be calculated using the formula , where is the speed measure in ft/sec and t is the time in seconds. Calculate the times when the speed is greater than 86 ft/sec.
Therefore when or , the speed is greater than 64 ft/sec.
Vocabulary
- Absolute Value Linear Inequality
- Absolute Value Linear inequalities can have one of four forms: , or . Absolute value linear inequalities have two related inequalities. For example for , the two related inequalities are and .
- Linear Inequality
- Linear inequalities can have one of four forms: , or . In other words, the left side no longer equals the right side, it is less than, greater than, less than or equal to, or greater than or equal to.
Guided Practice
1. Solve for the solution set to the inequality .
2. Solve for the solution set to the inequality .
3. Solve for the solution set to the inequality .
Answers:
1.
Therefore solution set is .
2.
Therefore solution set is .
3.
Therefore solution set is .
Practice
Find the solution sets for the variable in each of the following absolute value linear inequalities.
Find the solution sets for the variable in each of the following absolute value linear inequalities.
Find the solution sets for the variable in each of the following absolute value linear inequalities.
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Learning Objectives
Here you'll learn how to solve an absolute value inequality.
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Date Created:
Dec 19, 2012Last Modified:
Apr 29, 2014Vocabulary
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