Title: Algebra 1
Author: Mandy McDaniel
Last saved by: Glencoe Mathematics

Lesson 6-5

Example 1 Solve an Absolute Value Equation
a.  The temperature outside is 5°F give or take 6°F. Solve  = 6 to find the range of temperatures.
Method 1	Graphing
 = 6 means that the distance between s and 5 is 6 units. To find s on the number line, start at 5 and move 6 units in either direction.
								The distance from 5 to -1 is 6 units.

										The distance from 5 to 11 is 6 units.
The solution set is {-1, 11}.

Method 2	Compound Sentence
Write  = 6 as s – 5 = 6 or s – 5 = -6.
Case 1				Case 2
	s – 5 = 6					s – 5 = -6	
s – 5 + 5 = 6 + 5	Add 5 to each side.		s – 5 + 5 = -6 + 5	Add 5 to each side.
	s = 11	Simplify.		s = -1	Simplify.
The solution set is {-1, 11}. The range of temperatures is -1°F to 11 °F.

b.  Solve  = -5.
 = -5 means that the distance between s and 6 is –5. Since distance cannot be negative, the solution is the empty set, ∅.

Example 2 Write an Absolute Value Equation
Write an open sentence involving absolute value for the graph.

Find the point that is the same distance from 11 as the distance from 19. The midpoint between 11 and 19 is 15.
		The distance from 11 to 15 is 4 units.
	
				The distance from 19 to 15 is 4 units.
	So, an equation is  = 4.

Example 3	Solve an Absolute Value Inequality (<)
a.  Solve < 9. Then graph the solution set.
Write < 9 as 3p – 12 < 9 and 3p – 12 > -9.
Case 1	Case 2
	 3p – 12 < 9		 3p – 12 > -9	
	3p – 12 + 12 < 9 + 12	Add 12 to each side.	3p – 12 + 12 > -9 + 12	Add 12 to each side.	
		3p< 21	Simplify.	3p >  3	Simplify.
		 < 	Divide each side by 3.	>	Divide each side by 3.
		 p < 7	Simplify.	p > 1	Simplify.

The solution set is {p | 1  p  7}.	

b.  Solve  < -2.
Since  cannot be negative,  cannot be less than –2. So, the solution is the empty set, ∅.

Example 4 Solve an Absolute Value Inequality (>)
a.  Solve > 4. Then graph the solution set.
Write >4 as x – 7 4 and x – 7  -4.
Case 1  x – 7 is positive	Case 2  x – 7 is negative
					-(x – 7) > 4
	x – 7 >4		x – 7 < -4 	Divide each side by –1; 
reverse the symbol.
	x – 7 + 7 >4 + 7	Add 7 to each side.	x – 7 + 7 < -4 + 7	Add 7 to each side.	
		x > 11	Simplify.	x < 3	Simplify.
	
The solution set is {x | x  3 or x  11}.

b.  Solve  ≥ -4.
Since  is always greater than or equal to 0, the solution set is {x | x is a real number}.