Download M098 Carson Elementary and Intermediate Algebra 3e Section 2.6 Objectives

M098
Carson Elementary and Intermediate Algebra 3e
Section 2.6
Objectives
1.
2.
Represent solutions to inequalities graphically and using set notation.
Solve linear inequalities.
Vocabulary
Linear Inequality
An inequality containing expressions in which each variable term contains a single
variable with an exponent of 1.
Prior Knowledge
Inequality signs:
<
≤
>
≥
less than
less than or equal to
greater than
greater than or equal to
New Concepts
Represent solutions to inequalities graphically and using set notation.
Set-builder Notation: { variable | condition }
The | is called a pipe and is read “such that”.
{ x | x > -2 } is read “The set of all x such that x is greater than -2.” The condition is the most
important part of the notation.
Graphically: There are two methods to show if the endpoint of the graph is included in the set or not.
Open Circle/Closed Circle:
Use an open circle to indicate that the endpoint is not included in the set.
Use a closed circle to indicate that the endpoint is included in the set.
Parentheses ( and bracket [ method:
Use a parentheses to indicate that the endpoint is not included in the set.
Use a bracket to indicate that the endpoint is included in the set.
x<2
x≥2
V. Zabrocki 2011
or
or
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M098
Carson Elementary and Intermediate Algebra 3e
Section 2.6
Interval Notation: This is a shorthand method to represent multiple solutions. It uses the parenthesesbracket method discussed above. The smallest number in the set is written on the left and the largest
number is on the right. If the number is included (≤ or ≥), use brackets. If the number is not included (<
or >), use parentheses. We use ∞ to represent numbers that continue indefinitely in a positive direction.
We use -∞ to represent numbers that continue indefinitely in a negative direction.
Interval
Notation
Set builder
Notation
x<2
(-∞, 2)
{x|x<2}
x≥2
[2, ∞ )
{x|x≥2}
Graph
Compound or three-part inequalities.
A three part inequality like 4 < x < 9 is a way to show that the numbers we are interested in are between
4 and 9. Remember this does not mean just the whole numbers but all fractions between 4 and 9.
4<x<9
{x|4<x<9}
(4, 9)
Solve linear inequalities.
Solving inequalities is very similar to solving equations. There are just a couple of things to keep in mind:

With equations there was usually one answer, but with inequalities there are infinitely many
solutions. The answer is usually given in set-builder notation, graphically and in interval notation.

Reverse the direction of the inequality when we multiply or divide by a negative number.
Example 1:

5
p  10
3
 5 
3  p   3 10
 3 
-5p > -30
5p
30

5
5
There are 2 terms and the LCD is 3.
Multiply through by 3.
Reduce the fractions to eliminate the
denominators.
Divide both sides by -5. Remember to
switch the direction of the inequality.
p<6
Simplify the fraction.
{ p | p < 6}
Set-builder notation
Graph
, 6
V. Zabrocki 2011
Interval notation
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M098
Carson Elementary and Intermediate Algebra 3e
Section 2.6
Example 2:
1
1
6m  7  3m  1
5
2
 1
 1
10 6m  7  10 3m  1
5
2
There are 2 terms and the LCD is 10.
Multiply through by 10.
2(6m – 7) ≥ 5(3m - 1)
Reduce the fractions to eliminate the
denominators.
12m – 14 ≥ 15m – 5
Distribute to remove the parentheses.
-3m – 14 ≥ -5
Subtract 15m from both sides.
-3m ≥ 9
Add 14 to both sides
m ≤ -3
Divide by -3. Remember to reverse the
direction of the inequality.
{ m | m ≤ -3 }
Set-builder notation
Graph
,  3
V. Zabrocki 2011
Interval notation
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