Download 3-1 Inequalities and Their Graphs

3-1 Inequalities and Their Graphs
Hubarth
Algebra
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, –2 > 5 is not true.
b. 10
Yes, 10 > 5 is true.
c.
25
5
Yes, 5 > 5 is true.
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
3–4<8
–1 < 8
–2 is a solution.
Substitute for x.
Simplify.
Compare.
3 + 2(3) < 8
3+6<8
9<8
3 is not a solution.
Graphing and Writing Inequalities in One Variable
You can use a graph to indicate all of the solutions of an inequality.
Inequality
𝒙<𝟑
Graph
0
2
3
4
6
𝑚 ≥ −2
-4
−1 ≥ 𝑎
means
𝑎 ≤ −1
-3
-2
-1
0
1
The open circle shows that
3 is not a solution. Shade
all values to the left of 3
are solution also.
The closed circle shows that
-2 is a solution. Shade all
values to the right of -2 are
solutions also
The closed circle shows that
-1 is a solution. Shade all
values to the left of -1 are
solutions also
≤
}
≥
Closed circles because
they are equal to
<
}
≤
Shaded to the left
<
>}
Open circles because
they are not equal
>
}
≥
Shaded to the right
Ex 3 Graphing Inequalities
a. Graph d < 3.
b. Graph –3 ≥ g.
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 > 2
1
Numbers greater than
1
are graphed.
2
Practice
1. Is each number a solution of 𝑥 ≥ −4.1?
a. -5 No
b. -4.1
c. 8
Yes
Yes
d. 0
Yes
2. Is each number a solution of 6x – 3 > 10?
a. 1
b. 2
c. 3
No
No
Yes
d. 4
Yes
3. Graph each inequality.
a. 𝑎 < 1
-1
1
b. 2 ≥ 𝑝
3
0
2
𝑝≤2
4
4. Write an inequality for the graph.
x ≥ 10
8
10
12