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Math 142 — Rodriguez
Using Linear Inequalities in One Variable to Make Predictions
Lehmann — 3.5
I. Linear Inequalities in One Variable
A. The inequality symbols are <, ≤, >, and ≥ .
B. A linear inequality in one variable is a statement having one of the above inequality
symbols; only one variable is in it; and the exponent on the variable is a 1 (and so is
not written).
Before we look at how to solve linear inequalities, let’s look at how we will write the
answers.
II. Solution Set
A. The solution set is the set consisting of all the solutions to a given inequality.
B. Different ways to write the solution set:
Words
Inequality
Graph
Interval Notation
Numbers greater than 5
x≥1
(-∞,–7)
C. Recognize that we can ‘flip-flop’ the inequality and it is still a true statement.
Examples: 5 > 3
2<x
4>x
3<5
III.
Solving linear inequalities
A. The Addition Property of Inequality:
If a < b, then
a+c<b+c
and
a — c < b — c.
B. The Multiplication Property of Inequality:
If a < b and c is POSITIVE, then ac < bc
and
a
c
<
b
c .
If a < b and c is NEGATIVE, then ac > bc
and
a
c
>
b
c .
C. Solving linear inequalities is exactly the same as solving linear equalities EXCEPT when
multiplying or dividing both sides of the inequality by a negative number the
inequality symbol must be reversed.
Examples: Use a symbolic method to solve the inequality. Describe the solution set as an
inequality, in interval notation, and in a graph. Then, use graphing calculator tables or
graphs to verify your result.
x+4
2x − 1
+ 3 <
1) 3x − 1 ≤ 3 2x − 1 + 5
2)
6
4
(
)
(
) (
7
)
1
1
4) 6 x − 5 > 3 − 2 x
3) 3.5 + 0.2 2x − 1.4 ≤ 3 x + 2.3 − 0.04
IV. Solving Three-Part Inequalities
A. A three-part inequality is of the form a < bx + c < d. This is “shorthand” for
a < bx + c
Example:
and
bx + c < d
2<x≤8
2<x and x≤8
words:
graph:
interval notation:
B. To solve this type of inequality our goal is to isolate the variable in the middle part of
the three-part inequality. To do this, whatever we do to the middle part of the
inequality we do to the two outer parts of the inequality.
Examples: Use a symbolic method to solve the inequality. Describe the solution set as an
inequality, in interval notation, and in a graph. Then, use graphing calculator tables or
graphs to verify your result.
1)
3 ≤ 2x + 4 < 12
Lehmann — 3.5
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2)
3
2
< 4− x ≤2
5
5
3)
−8 ≤ 3x − 5 < 11
4)
−7 < 3− 2x ≤ 11
IV. Applications involving inequalities
These applications use phrases like ‘more than’ or ‘less than’ to compare quantities. You
will have to write an inequality to express the information given in the application. Your
answer will be a phrase/sentence that describes the solution set.
Equations given:
Lehmann — 3.5
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Find equations given slopes (rates of change) and intercepts (starting values):
Find equations given table with data:
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