Simple Inequalities
When we examine an inequality, we are searching for the values of x that will make the
inequality true.
Ex: x  2
What values of x make this sentence true?
Four types of inequalities:
1. Greater than: >
2. Less than: <
3. Greater than or equal to: 
4. Less than or equal to: 
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Sketch the following:
A. x < 7
B. x  2
C. x > 2
D. x  5
Q: How can we sketch the graph of x  3  5 ?
How can we make this inequality look like an inequality we know how to graph?
We can solve linear inequalities much like we solve linear equations. Isolate the variable
on one side of the inequality and graph as usual.
Ex:
x 3 5
+3 +3
x < 8
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Types of transformations:
1. Add the same number to both sides.
2. Subtract the same number from both sides.
x 3 5
x  6  10
1
x3
3. Multiply both sides by the same positive number.
2
4. Divide both sides by the same positive number.
3x  9
5. Multiply both sides by the same negative number and reverse the inequality.
x4
6. Divide both sides by the same negative number and reverse the inequality.
 2x  6
ONE-STEP:
Ex:
p53
-5 -5
p  2
-----------------------------
Examples: Solve and Graph the following.
1) x  5  7
2) 4  x  12
3)  8  x  17
4)  3  x  19
5) 6  x  18
6) x  10  6
MULTI-STEP:
Ex:
2y  5  7
+5 +5
2 y  12
2
2
y < 6
-----------------------------
Examples: Solve and Graph the following.
1) 2x  7  4
2) 6  3x  15
3)  4x  2  10
4) 2x  5  13
5)  3  6x  1
6) 17  4x  11
REVERSING INEQUALITIES:
Ex:
5 x  4
-5
-5
 x  1
-1 -1
x 1
Examples: Solve and Graph the following.
1) 6  x  12
2)  x  5  25
VARIABLES ON BOTH SIDES:
Hint: Combine like terms!
Remember transformations 5 and 6!
You can multiply or divide by -1. SAME!
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3)  3  x  81
Ex:
2x  4  4x  1
+4
+4
2x  4x  3
-4x -4x
 2x  3
-2
-2
2
x
3
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Examples: Solve and Graph the following.
1)  x  4  3x  2
2) 3x  5  7 x  9
3) x  3  2( x  4)
4) 2 x  10  7( x  1)
5)  x  4  2( x  8)
6)  x  6  (2 x  4)
Try some word problems!
1. The lowest elevation in California is 282 below sea level. Let E represent the
elevation of any location in California. Write an inequality for E and graph.
2. During 1993, each monthly company profit, P, was at least $1200. Write an
inequality for this statement and graph.
Homework: Work Book
p. 68, 27-42 EVEN
p. 70, 26-45 EVEN
p. 72, 1-18 ALL for multi-step