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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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