# Inequalities

**Overview
**• Section 1.5 in the textbook:

– Linear Inequalities

– Compound Inequalities

– Absolute Value Inequalities

– Polynomial Inequalities

– Rational Inequalities

**Linear Inequalities**

** Solving Inequalities
**• An inequality is similar to an equality except

instead of =, we have >, <, ≤ or ≥.

– Essentially solved in the same way as an equality

• Solution set – all values that satisfy an

inequality

• Whereas an equality has at most 1 solution, the

solution to an inequality is a set – possibly with

infinitely many elements

• You will have an easier time solving inequalities

if the variable is isolated on the LEFT side

**Dividing an Inequality by a
Negative
**• Only point where solving an inequality

differs from solving an equality

• If the inequality is DIVIDED (or

MULTIPLIED ) by a NEGATIVE number,

SWITCH the direction of the inequality

• Adding negative numbers to both sides of

an inequality does NOT switch the

direction of the inequality!

**Review of Graphing & Interval
Notation
**• When graphing the solution set on a number

line:

– If the inequality is > or <, use parentheses ( )

– If the inequality is ≥ or ≤, use brackets [ ]

• Interval notation represents the “endpoints” of

the graph of the number line

– First value is what is shaded furthest to the left on

the graph

– Second value is what is shaded furthest to the right

on the graph

• A shaded arrow on the number line represents ∞

– Parentheses ALWAYS go around infinity

**Solving Inequalities (Example)
**Ex 1: Solve, graph, and write the solution in

interval notation:

5(x - 6)+ 4 ≤ 5x - 2(1- x) |

Ex 2: Solve, graph, and write the solution in

interval notation:

**Compound Inequalities**

**Solving Compound Inequalities
Using Union
**• Two inequalities separated by the word or

• Solve each linear inequality as normal

• To graph, draw 3 number lines with equal

intervals:

– On the first number line, graph the solution to

the first inequality

– On the second number line, graph the

solution to the second inequality

– On the third number line, lay the first two

number lines on top of each other – this

represents the union

• Determine the endpoints and remove any

parentheses or brackets that are not on the

endpoints

– Obtain the interval notation from the union

• If two areas of the number line are

shaded:

–Write the interval notation for the left part

–Write the interval notation for the right part

– Union the two intervals

Ex 3: Solve, graph, and write the solution in

interval notation:

5x + 17 ≥ 22 or 2(x – 3) + 1 < -13

Ex 4: Solve, graph, and write the solution in

interval notation:

3(x – 4) + 8 > 5 or 10x ≤ 30

Ex 5: Graph and write the solution in

interval notation:

x ≥ 2 or x > 7

**Compound Inequalities Separated
by and
**• Solve each linear inequality as normal

• To graph, draw 3 number lines with equal

intervals:

– On the first number line, graph the solution to

the first inequality

– On the second number line, graph the

solution to the second inequality

– On the third number line, lay the first two

number lines on top of each other

– The intersection is the area between the left

parenthesis (or [) and the right parenthesis

(or ])

• Obtain the interval notation from the intersection

– It is possible that there is no intersection

Ex 6: Solve, graph, and write the solution in

interval notation:

2(x – 3) < -9 and -2x + 7 ≤ 3x + 22

Ex 7: Solve, graph, and write the solution in

interval notation:

5(x – 2) < -10 and -9x < -18

**Compound Inequalities with Two
Inequality Symbols
**• Most common way to see a compound inequality

involving intersection

– Ex: -2 < x – 3 < 5

• To solve:

– Goal is to isolate the variable between the two

inequality symbols

– Perform Algebraic operations on three sides instead

of two

• Simple to graph:

– Once the variable is isolated, the intersection is

already obtained

• No need to draw three graphs

Ex 8: Solve, graph, and write the solution in

interval notation:

-12 ≤ 2 – 5x < 7

Ex 9: Solve, graph, and write the solution in

interval notation:

**Absolute Value Inequalities**

**Absolute Value Inequalities
**• Recall that absolute value measures distance

• If we say |x| < c, then the solution set contains all points

c units from 0 (left and right)

– Graph:

– Interval notation: (-c, c)

• If we say |x| > c, then the solution set contains all points

more than c units from 0 (left and right)

– Graph:

– Interval notation: (-oo, c) U (c, +oo)

**Solving Absolute Value Inequalities
**• The absolute value inequality can be

transformed into a compound inequality based

on the inequality sign:

– If < or ≤,

• Intersection

– If > or ≥,

• Union

Ex 10: Solve, graph, and write the solution

in interval notation:

|3x – 2| – 2 > -1

Ex 11: Solve, graph, and write the solution in

interval notation:

**Polynomial Inequalities**

**Solving Polynomial Inequalities
**• We discussed how to solve polynomial

equalities

• Only difference is that the solution set now

consists of intervals instead of real

numbers

• To solve an inequality such as

(x + 1)(x – 1) > 0:

– By the Zero Product Principle, x = -1 or x = 1

– This subdivides the interval (-oo, +oo) into

three subintervals:

(-oo, -1), (-1, 1), (1, +oo)

– Sign Property of polynomials : if one value

in a subinterval yields a certain sign, then

ALL values in the subinterval have that SAME

sign

• First must separate (-oo, +oo) into subintervals

by using the solutions to the polynomial

• Ex: Since x = -5 24 > 0, ALL values in (-oo, -1)

are positive

– Thus, pick one value in each subinterval and

test it in the inequality

• Keep only those intervals that satisfy the inequality

• If more than one interval satisfies the inequality,

union them

Ex 12: Solve and write the solution in

interval notation:

2x^{2} – x < 10

Ex 13: Solve and write the solution in

interval notation:

x^{3} + x^{2} – 12x ≥ 0

**Rational Inequalities**

**Solving Rational Inequalities
**• Almost the same process as solving a

polynomial inequality

• Also need to consider values for the

variable that cause the denominator to

equal 0

• To solve a rational inequality:

– Set one side to ZERO and write the other side

as a rational expression

– Determine the critical values – those values

that cause either the numerator or

denominator to equal 0

– Split (-oo, +oo) into subintervals based on

the critical values

• Just as with a polynomial inequality

– Test each subinterval to determine those that

are included in the solution set

• The sign rule for polynomials applies for rational

expressions as well

– If necessary, adjust the intervals for

extraneous solutions!

Ex 14: Solve and write the solution in interval

notation:

Ex 15: Solve and write the solution in interval

notation:

** Summary
**• After studying these slides, you should know how to do

the following:

– Solve, graph, and write the solution set in both interval and set

builder notation for the following types of inequalities:

• Linear

• Compound

• Absolute value

• Polynomial

• Rational

• Additional Practice

– See the list of suggested problems for 1.5

• Next lesson

– Graphing & Circles (Section 2.1)

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