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EXAMPLES & SOLUTIONS
The value substituted for the inequality variable must provide a correct mathematical statement.
The inequality values on one side is less than or greater than the other side.
Some values maybe negative and may need to be adjusted.
The adjustments are needed for a correct mathematical statement.
INEQUALITY EXPRESSIONS
Explanation of evaluating expressions
Fact: A expression on one side is either less than; greater than; less than & equal; or greater than & equal to another expression.
expression: x > 3
All numbers > 3 are correct for this expression
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Inequality Expressions
Examples of evaluating expressions
expression: 7 + x < 10
if x = 3
Substitute 3 for x:
7 + 3 < 10
10 < 10 --correct
Addition
if x = 3, evaluate: 10 + x > 4
10 + x > 4 --substitute 3 for x
10 + 3 > 4 --addition rule
13 > 4 --correct
Multiply
if x = 3, evaluate: 10 * x < 4
4 < 10 * x --substitute 3 for x
4 < 10 * 3 --multiplication rule
4 < 30 --correct
Subtraction
if x = 3, evaluate: 10 - x > 4
10 - x > 4 --substitute 3 for x
10 - 3 > 4 --subtract rule
7 > 4 --correct
Divide
if x = 3, evaluate: 10 < 4
x
10 < 4
10 < 4
3.333 < 4 --correct
3
x
-- substitute 3 for x
-- division rule
Sign Reversal
Explanation of the reversal of signs
Fact: Multiplying or dividing any real neg. number on both sides, requires that the sign to be reversed.
expression: -2n - 3n < 30
Combine terms:
-2n - 3n < 30 --subtract rule
Updated Expression:
-5n < 30
Isolate n:
-5n < 30
(-5)
(-5)
Solution:
n < -6
This equation states that all n-values must be
less than -6
Is this the correct sign for this inequality? NO!!
WHY??
Choose any number less than -6:
example: x = -7
Evaluate with original expression:
-5n < 30
-5 (-7) < 30 --multiplication rule
35 < 30 -- NOT CORRECT
Remember, switch the sign whenever you multiply or divide by a neg. number.
SWITCH THE SIGNS
n > -6
This equation states that all n-values must be
greater than -6
Is this the correct sign for this inequality? YES!!
WHY??
Choose any number greater than -6:
example: x = 7
Evaluate with original expression:
-5n < 30
-5 (7) < 30--multiplication rule
-35 < 30-- CORRECT!!!
Inequality Expressions: Sign Reversal
Examples of the reversal of signs
Multiplication
if x = -24, evaluate: x < 5
-4
-4
x < 5 --multiply each side by -4
4x < 5 (-4)
(-4)
x < -20 --updated expression
--CHECK: PLUG IN A NUMBER LESS THAN -20
6 < 5 --NOT CORRECT (switch sides)
x > -20 --CORRECT EXPRESSION
--CHECK: PLUG IN A NUMBER GREATER THAN -20
24 < 5 --substitute x = 24
-6 < 5 --CORRECT
-24 < 5 --substitute x = -24
-4
-4
Division
evaluate: -7 * x > 14
-7
-7 * x > 14 --multiplication rule
-7x > 14 --divide both sides by -7
-7
x > -2 --updated expression
--CHECK: PLUG IN A NUMBER GREATER THAN -2
-3 > 14 --NOT CORRECT (switch sides)
x < -2 --CORRECT EXPRESSION
--CHECK: PLUG IN A NUMBER LESS THAN -2
-21 < 14 --substitute x = 24
3 < 14 --CORRECT
21 > 14 --substitute x = -21
-7
-7
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