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SOLVING INEQUALITIES
Chapter 3
Introduction
• In this chapter we will extend the skills learned in
the previous chapter for solving equalities to
inequalities.
• Many of the procedures used will be the same
reflecting that the properties of inequalities are
very similar to those for equations.
• Some key difference exists though.
• We will also learn to solve for equations and
inequalities involving absolute values.
Inequalities and Their Graphs (3.1)
Solution of an inequality: Any number that
makes the inequality true.
• When graphing solutions of inequalities:
• Open dot: The particular number is not part of the
solution set.
• Closed dot: The particular number is a part of the
solution set.
Solving Inequalities Using Addition and
Subtraction (3.2)
Equivalent inequalities: Inequalities with the same
solutions.
• We can add or subtract the same value to each side of an
inequality just as we did with equations to solve for an
inequality.
Addition Property of Inequality
For every real number a, b, or c,
If a > b, then a+c > b+c
If a<b, then a+c<b+c
Subtraction Property of Inequality
For every real number a, b, or c,
If a > b, then a-c > b-c
If a<b, then a-c<b-c
Solving Inequalities Using Addition and
Subtraction (3.2)
Sample Problem
a) Solve 𝑥 − 3 < 5
b) Solve 12 ≤ 𝑥 − 5
c) Solve 𝑦 + 5 < −7
Solving Inequalities Using Multiplication
and Division (3.3)
• We can multiply both sides of an inequality in much the
same way we did with an equation.
• There is one key exception and it has to do with whether
we are multiplying by a negative or a positive number:
When multiplying by
a positive number.
Do not change the
direction of the inequality
symbol.
When multiplying by
a negative number.
Change the
direction of the inequality
symbol.
Solving Inequalities Using Multiplication
and Division (3.3)
There are two properties that deal with multiplying
inequalities:
Multiplication Property of Inequality for c >0
(c is a positive number)
For every real number a, b, or c,
If a > b, then ac > bc If a<b, then ac<bc
Multiplication Property of Inequality for c < 0
(c is a negative number)
For every real number a, b, or c,
If a > b, then ac < bc If a<b, then ac>bc
Solving Inequalities Using Multiplication
and Division (3.3)
Sample Problem
a) Solve:
b) Solve:
𝑥
2
< −1
2
− 𝑛
3
≤2
Solving Inequalities Using Multiplication
and Division (3.3)
There are two properties that deal with dividing inequalities:
Division Property of Inequality for c >0
(c is a positive number)
For every real number a, b, or c,
a
b
a
b
If a > b, then >
If a<b, then <
c
c
c
c
Division Property of Inequality for c < 0
(c is a negative number)
For every real number a, b, or c,
a
b
a
b
If a > b, then <
If a<b, then >
c
c
c
c
Solving Inequalities Using Multiplication
and Division (3.3)
Sample Problem
a) Solve: −5𝑧 ≥ 25
b) Solve: 13.75 × 𝑐 ≤ 216
Solving Multi-Step Inequalities(3.4)
• Solving multi-step inequalities is the same as solving
multi-step equations.
• Simplify by dealing with the addition or subtraction first by carrying
out the inverse operation.
• Then simplify the multiplication/division by again by taking the
inverse operation.
• Remember, simply multiplying by the inverse of the coefficient will be
the same as applying the inverse operation.
• Again the exception occurs when we multiply or divide by
a negative number to simplify the inequality:
• When multiplying/dividing by a negative number remember to
reverse the inequality sign.
Solving Multi-Step Inequalities(3.4)
Sample Problem
a) Solve 7 + 6𝑎 > 19
b) Solve 36 + 2𝑤 ≤ 48
Solving Multi-Step Inequalities(3.4)
Sample Problem
a) Solve 2 𝑡 + 2 − 3𝑡 ≥ −1
Solving Multi-Step Inequalities(3.4)
Sample Problem
a) Solve 6𝑧 − 15 < 4𝑧 + 11
b) Solve −3 4 − 𝑚 ≥ 4(2𝑚 + 1)
Compound Inequalities (3.5)
Compound inequality: Two inequalities that are joined by
the word “and” or the word “or”.
Example 1
𝑥 > −4 𝑎𝑛𝑑 𝑥 ≤ 3. This can also be written as − 4 < 𝑥 ≤ 3
• The solution to compound inequalities such as the one
above is in the overlap of the solutions of the individual
parts of the compound inequality.
Compound Inequalities (3.5)
• Solving a compound inequality containing
the word “and.”
• The solution involves solving all sides the same
way.
• What is done to one side of the compound inequality must be
done to all sides of the compound inequality.
• Again, remember when multiplying or dividing
by a negative number the inequality signs must
be reversed.
Compound Inequalities (3.5)
Sample Problem
Solve −4 < 𝑟 − 5 ≤ −1. Graph the solution.
Compound Inequalities (3.5)
Example 2
𝑥 ≥ 3 𝑜𝑟 𝑥 < −4.
• The solution to compound inequalities such as the one
above shows that there is no overlap of the solutions of
the individual parts of the compound inequality.
• When solving for compound inequalities with “or,” solve
for the each inequality separately.
• In solving for each inequality, use the same rules for solving
inequalities learned above.
Compound Inequalities (3.5)
Sample Problem
Solve 4𝑣 + 3 < −5 𝑜𝑟 − 2𝑣 + 7 < 1. Graph
the solution.
Absolute Value Equations and Inequalities
(3.6)
• Remember, the absolute value of a number is its distance
from zero. Therefore, the absolute value can never be
negative.
• In absolute value equations, the unknowns will be set to
both positive and negative values..
Solving Absolute Value Equations
To solve an equation in the form │A│= b, where
A represents a variable expression and b > 0,
solve A =b and A=−𝑏.
Absolute Value Equations and Inequalities
(3.6)
Sample Problems
a) Solve 𝑥 + 5 = 11
b) Solve 2𝑝 + 5 = 11
Absolute Value Equations and Inequalities
(3.6)
• Absolute value inequalities also exist.
• When solving these absolute value inequalities,
write the inequalities as compound inequalities.
Solving Absolute Value Inequalities
To solve an inequality in the form │A│< b, where
A represents a variable expression and b > 0,
solve −𝑏 < 𝐴 < 𝑏.
To solve an inequality in the form │A│> b, where
A represents a variable expression and b > 0,
solve 𝐴 < −𝑏 𝑜𝑟 𝐴 > 𝑏.
Absolute Value Equations and Inequalities
(3.6)
Sample Problems
a) Solve 𝑣 − 3 ≥ 4
THE END
SOLVING INEQUALITIES
Chapter 3