Equations and
Inequalities
A-6
Solve Linear Inequalities
Objectives
Solve and Graph Linear Inequalities
 Solve and Graph Compound Inequalities

Essential Question
How are the rules for solving linear
inequalities similar to those for
solving linear equations, and how are
they different?
A linear inequality is a statement that
compares two expressions by using the
symbols <, >, ≤, ≥, or ≠. A solution of an
inequality is the set of all points that satisfy
the inequality and makes it a true statement
when you substitute the number(s) for the
variable. The graph of an inequality consists
of all points on a number line that represent
solutions. The properties of equality are true
for inequalities, with one important difference.
If you multiply or divide both sides by a
negative number, you must reverse the
inequality symbol.
REMEMBER!
Use an open circle with an arrow to the
left for < and an arrow to the right for >.
Use a closed circle (solid dot) with an
arrow to the left for ≤ and an arrow to the
right for ≥.
Solve
Graph the solution set on a
number line. Answer using interval notation.
Original inequality
Add –4y to each side.
Simplify.
Add –2 to each side.
Simplify.
Rewrite with y first using
symmetry.
Answer: Any real number greater than –5 is a solution
of this inequality.
An open circle means that
Interval notation: (−5, ∞)
this point is not included in
the solution set.
Solve
Graph the solution set on a
number line. Answer in interval notation.
Answer: (−∞, 𝟗)
Multiplying or dividing each side of an
inequality by a positive number does
not change the truth of the inequality.
However, multiplying or dividing each
side of an inequality by a negative
number requires that the order of the
inequality be reversed. For example, to
reverse ≤, replace it with ≥.
Solve
Graph the solution set on a
number line. Answer in set-builder notation.
Original inequality
Divide each side by –0.3,
reversing the inequality symbol.
–40  p
p  –40
Simplify.
Rewrite with p first.
Answer: The solution set is
A solid dot means that
this point is included in
the solution set.
Solve
Graph the solution set on a
number line. Write your answer using set-builder
notation.
Answer:
Solve
Graph the solution set on a
number line. Write your answer using interval
notation.
Original inequality
Multiply each side by 2.
Add –x to each side.
Divide each side by –3,
reversing the inequality symbol.
Answer: The solution set is
graphed below.
and is
Solve
Graph the solution set on a
number line. Write your answer in interval notation.
Answer:
A compound inequality consists of two
inequalities joined by the word and or the
word or. To solve a compound inequality, you
must solve each part of the inequality.
A compound inequality containing the word
and is true if and only if both inequalities are
true.
A compound inequality containing the word
or is true if one or more of the inequalities is
true.
“And” Compound Inequalities
The graph of a compound inequality
containing and is the intersection of the
solution sets of the two inequalities. The
mathematical symbol for intersection is ∩.
Compound inequalities involving the word and
are called conjunctions.
Solution Notations
The most common way of writing the solution
x ≥ –1 and x < 2 is –1 ≤ x < 2. Both forms
are read “x is greater than or equal to –1 and
less than 2”. Both inequality symbols for
conjunctions are always < or ≤.
In set-builder notation: {x | –1 ≤ x < 2 }
In interval notation: [–1, 2 ), indicating that
the solution set is the set of all numbers
between –1 and 2, including –1, but not
including 2.
Solve
Graph the solution set on a number line.
Method 1 Write the compound inequality using the word
and. Then solve each inequality.
and
Method 2 Solve both parts at the same time by adding 2
to each part. Then divide each part by 3.
12 ≤
Graph the solution set for each inequality and find
their intersection.
y4
Answer: Set Builder: {y | 4 ≤ y < 7}
Interval Notation: [ 4, 7)
Solve
Graph the solution set on a number line. Write
your answer in set-builder notation.
Answer:
“Or” Compound Inequalities
The graph of a compound inequality
containing or is the union of the solution
sets of the two inequalities. The
mathematical symbol for union is U.
Compound inequalities involving the word or
are called disjunctions.
Solution Notations
The most common way of writing x ≤ 1 or x > 4 is
by using the word “or” between the two inequalities.
In set-builder notation: {x | x ≤1 or x > 4}
In interval notation, the mathematical symbol for the
union of the two sets is used and written as:
(–∞, 1] U (4, +∞), indicating that all values less
than and including 1 are part of the solution set. In
addition, all values greater than 4, not including 4,
are part of the solution set.
Solve
or
Graph the solution set on a number line. Write the
solution in set-builder notation.
Solve each inequality separately.
or
Answer: The solution set is
Solve 𝒙 + 𝟓 < 𝟏 𝒐𝒓 − 𝟐𝒙 ≤ −𝟔.
Graph the solution set on a number line. Write the
answer in set-builder notation.
Answer:
Solutions to Inequalities
If you arrive at a false statement, such as 3 > 5,
then the solution set for that inequality is the
empty set { } or null set Ø.
 If you arrive at a true statement such as 3 > –1,
then the solution set for that inequality is the set
of all real numbers, written in set builder
notation as {x | x Є R}, read as “ the set of all
x ‘s such that x is an element of the real number
system”.

Inequality Phrases
< is less than;
is fewer than
 > is greater than;
is more than; or
better than

≤ is at most;
is no more than;
is less than or
equal to
 ≥ is at least;
is no less than;
is greater than or
equal to

Essential Question
How are the rules for solving linear
inequalities similar to those for solving linear
equations, and how are they different?
The addition and subtraction properties are
the same, but if you multiply or divide both
sides of an inequality by a negative number,
the inequality symbol must be reversed.
Math Fact!

The cruise ship Queen Elizabeth 2 travels
40 feet on 1 gallon of fuel.