1. 3-1 Inequalities and Their Graphs
Hubarth
Algebra

2. A solution of an inequality is any number that makes the inequality true. For example,
the solution of the inequality 𝑥<3 are all numbers less than 3. So 2, 1, 0, -1,… etc. would
be solutions.
Ex 1 Identifying Solutions by Mental Math
Is each number a solution of x 5?
>
a. –2
No, – is not true.
>
b. 10
Yes, is true.
> 25 5 c.
Yes, is true.
>

3. Ex 2 Identifying Solutions by Evaluating
Is each number a solution of 3 + 2x < 8?
a. –2
b. 3
3 + 2x < 8
3 + 2x < 8
3 + 2(–2) < 8
Substitute for x.
3 + 2(3) < 8
3 – 4 < 8
Simplify.
3 + 6 < 8
–1 < 8
Compare.
9 < 8
–2 is a solution.
3 is not a solution.

4. Graphing and Writing Inequalities in One Variable
You can use a graph to indicate all of the solutions of an inequality.
Inequality Graph
𝒙<𝟑
The open circle shows that
3 is not a solution. Shade
all values to the left of 3
are solution also. 2 3 4 6
The closed circle shows that
-2 is a solution. Shade all
values to the right of -2 are
solutions also
𝑚≥−2 -4 -2
The closed circle shows that
-1 is a solution. Shade all
values to the left of -1 are
solutions also
−1≥𝑎
means
𝑎≤−1 -3 -1 1

5. } } } } ≤ < ≥ ≤ < > > ≥ Closed circles because
they are equal to
Shaded to the left ≥ ≤
<
> } }
Open circles because
they are not equal
Shaded to the right
> ≥

6. Ex 3 Graphing Inequalities
a. Graph d < 3.
b. Graph –3 ≥ g.

7. Ex 4 Writing an Inequality from a Graph
Write an inequality for each graph. a.
x < 2 Numbers less than 2 are graphed. b.
x –3 Numbers less than or equal to –3 are graphed.
< c.
x > –2 Numbers greater than –2 are graphed. d.
x > Numbers greater than are graphed. 1 2

8. Practice
1. Is each number a solution of 𝑥≥−4.1?
a. -5 b c. 8 d. 0 No
Yes
Yes
Yes
2. Is each number a solution of 6x – 3 > 10?
a. 1 b. 2 c. 3 d. 4 No No
Yes
Yes
3. Graph each inequality.
a. 𝑎<1 b. 2≥𝑝
𝑝≤2 -1 1 3 2 4
4. Write an inequality for the graph.
x ≥10 8 10 12