What you should learn

To solve open sentences involving absolute value and graph the solutions

NCTM Curriculm Standards 2, 6 - 10

 

Introduction: On Tuesday, February7, 1995, the space shuttle Discovery maneuvered within 37 feet of a Russian space station Mir, 245 miles above Earth. To accomplish this feat, Discovery had to be launched within 2.5 minutes of a designated time (12:45 AM). This time period is known as the launch window. If t represents the time elapsed since the launch countdown began and there are 300 minutes scheduled from the beginning of the countdown to blast-off, you can write the followign inequality to represent the launch window.

|300 - t| 2.5

We use absolute value because 300 - t cannot be negative.

There are three types of open sentences that can involve absolute value. They are as follows, when n is nonnegative.

First let's consider the case of |x| = n.

If |x| = 4, this means that the difference from 0 to x is 4 units.

Therefore, if |x| = 4, then x = -4 or x = 4. The solution set is {-4, 4}. So, if |x| = n, then x = -n or x = n.

Equations involving absolute value can be solved by graphing them on a number line or by writing them as a coupound sentence and solving it.

Exercise 1: Solve |x - 3| = 5

Now let's consider the case of |x| < n. Inequalities involving absolute value can also be represented on a number line or as compound inequalities. Let's examine |x| < 4.

|x| < 4 means that the distance from 0 to x is less than 4 units.

Therefore, x > -4 and x < 4. The solution set is {x|-4 < x < 4}. So if |x| < n, then x > -n and x < n.

Exercise 2: Solve |3 + 2x| < 11 and graph the solution set.

Finally let's examine |x| > 4. This means that the distance from 0 to x is greater than 4 units.

Therefore, x < -4 or x > 4. The solution set is {x|x < -4 or x > 4}. So, if |x| > n, then x < -n or x > n

Exercise 3: Solve |5 + 2y| 3 and graph the solution set.

Organizations such as OSHA set standards for buildings to meet the needs of those using the building. Building code standards are often written as maximums or minimums that must be met. These standards can often be written as inequalities.

Exercise 4: Many specifications in the building industry address the needs of physically-challenged persons. For example, hallwyas in hospitals must have handrails. The handrails must be places within a range of 2 inches from a height of 36 inches.

a. Write an open sentence that involves absolute value to represent the range of acceptable heights for hallway handrails.

b. Find and graph the corresponding compound sentence.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 17 - 55 odd, 57 - 66

Alternative Homework: Enriched: 18 - 46 even, 48 - 66

Extra Practice: Students book page 773 Lesson 7-6

Extra Practice Worksheet: Click Here.

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