Download I. Linear Inequalities in one variable

Math 154 — Rodriguez
Angel — 2.8
Inequalities in One Variable
I.
Linear Inequalities in one variable
A. The inequality symbols are <, ≤, >, and ≥ .
B. These symbols can be use to compare numbers. Examples: 3 > −5 or 4 ≥ 4.
C. In this section we look at linear inequalities in one variable. 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; that is you see x not x1).
Examples: 1) x > 5
2) 2x − 4 ≤ 32
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. The simplest inequality looks like x > 5 . In words, this says “all numbers that are greater than 5”.
We can’t list all numbers that are greater than 5 so instead we show the solution set on a graph.
Inequality
x>5
x≥ 5
x<5
x≤5
Graph
5
5
5
5
C. Recognize that we can ‘flip-flop’ the inequality and it is still equivalent.
Examples:
5>3
x>2
x<4
3<5
2<x
4>x
III. Solving linear inequalities
A. The Addition Property of Inequality states that if the same amount is added to (or subtracted
from) both sides of an inequality the resulting inequality is equivalent to the original one.
Example:
5>3
5+4 > 3+ 4
9 > 7 still a true statement
5>3
5−4>3−4
1 > −1 still a true statement
B. The Multiplication Property of Inequality states that if both sides of an inequality are multiplied
(or divided) by the same POSITIVE number, the resulting inequality is equivalent to the original
one
AND
that if both sides of an inequality are multiplied (or divided) by the same NEGATIVE number, the
inequality symbol must be REVERSED in order for the resulting inequality to be equivalent to the
original one.
Example: 6 > 3
6 ⋅ 2 > 3⋅ 2
12 > 6 still true
Example:
6>3
True statement
6
3
>
−3 −3
Divided by a negative number
−2 > −1
−2 < −1
IF we keep the inequality symbol as is, we get a false statement.
IF we REVERSE the inequality symbol, it is a true statement.
C. Solving linear inequalities is exactly the same as solving linear equalities (sections 2.1−2.5)
EXCEPT when multiplying or dividing both sides of the inequality by a negative number the
inequality symbol must be reversed.
D. Steps:
1)
2)
3)
4)
5)
6)
If fractions are present, multiply both sides of the inequality by the LCD (least common
denominator). After you are done multiplying by the LCD the inequality will no longer have
fractions.
Use the distributive property to remove parentheses.
Combine like terms that are on the SAME side of the inequality symbol.
Use the Addition Property to move all variable terms to the one side of the inequality and all
constant terms (numbers) to the other side of the inequality. By the time you are done with
this step, the equation will look like ax > b (or any of the other inequality symbols) or the
variables will cancel out*.
*If the variables cancel out, look at what you are left with.
If you are left with a TRUE statement, then ALL REAL NUMBERS are solutions to inequality.
If you are left with a FALSE statement, then there is NO SOLUTION to the inequality.
Use the Multiplication Property to isolate the variable; that is, to get it to look like x > # (any
of the inequality symbols). Remember to reverse the inequality symbol if you multiply or
divide both sides of the inequality by a negative number.
Graph the solution set.
Examples: Solve each inequality. Graph the solution set on a number line.
1) x − 6 > 9
2) −3x + 5 ≤ 11
3) 5(3 − x) ≤ 3x − 1
4) 2(x − 4) + 5 ≤ −5x + 8 + 7x
Angel — 2.8
Page 2 of 3
5)
3x
1 x
+1> −
10
5 10
6)
3
2
(n − 4 ) ≤ (n − 4 )
4
3
Now you try:
7) 5 ( x − 4 ) > 4 ( x + 8 )
8) −5x + 6 < 8 − 4x
9) 5x − 6 > 9 + 5x
10)
Angel — 2.8
x
x
− 5≥
6
2
Page 3 of 3