Solving Inequalities Part 2 Lesson

In the last lesson, we learned the basics of how to solve a linear inequality in one variable. In that lesson, we learned about interval notation, how to graph an interval on the number line, the addition property of inequality, and the multiplication property of inequality. It is very important to understand these concepts before moving forward. In this lesson, we will focus on a general procedure to solve any linear inequality in one variable.

Solving a Linear Inequality in One Variable

1. Simplify each side completely
   • Just like with equations, we can clear decimals or fractions
2. Isolate the variable term on one side of the inequality
   • We perform this action using the addition property of inequality
3. Isolate the variable
   • We perform this action using the multiplication property of inequality

Let's jump in and look at a few examples.
Example 1: Solve each inequality, write in interval notation, graph.
31 + 8x ≥ 5(3x - 5)
1) Simplify each side
31 + 8x ≥ 15x - 25
2) Isolate the variable term on one side
31 - 31 + 8x ≥ 15x - 25 - 31
8x ≥ 15x - 56
8x - 15x ≥ 15x - 15x - 56
-7x ≥ -56
3) Isolate the variable
-7/-7 x ≥ -56/-7
x ≤ 8
Remember, if we divide by a negative, we flip the direction of the inequality symbol.
Interval Notation:
(-∞,8]
Graphing the Interval: Example 2: Solve each inequality, write in interval notation, graph.
7(x - 8) - 8x > -8 + 7x
1) Simplify each side
7x - 56 - 8x > -8 + 7x
-x - 56 > -8 + 7x
2) Isolate the variable term on one side
-x - 56 + 56 > -8 + 56 + 7x
-x > 48 + 7x
-x - 7x > 48 + 7x - 7x
-8x > 48
3) Isolate the variable
-8/-8 x > 48/-8
x < -6
Remember, if we divide by a negative, we flip the direction of the inequality symbol.
Interval Notation:
(-∞,-6)
Graphing the Interval: Example 3: Solve each inequality, write in interval notation, graph.
-9(1 + 8x) - 4x ≥ - 10(9x + 9) + 11
1) Simplify each side
-9 - 72x - 4x ≥ -90x - 90 + 11
-9 - 76x ≥ -90x - 79
2) Isolate the variable term on one side
-9 + 9 - 76x ≥ -90x - 79 + 9
-76x ≥ -90x - 70
-76x + 90x ≥ -90x + 90x - 70
14x ≥ -70
3) Isolate the variable
14/14 x ≥ -70/14
x ≥ -5
Interval Notation:
[-5,∞)
Graphing the Interval:

Solving a Three-Part Inequality

We will also come across three-part inequalities. To solve these, we just follow a similar set of steps. Our goal is to isolate the variable in the middle. We will perform the same action to each part until we accomplish our goal of:
some number < x < some number
Let's look at a few examples.
Example 4: Solve each inequality, write in interval notation, graph.
-7 ≤ x - 1 ≤ 9
To isolate x in the middle, let's add 1 to each part:
-7 + 1 ≤ x - 1 + 1 ≤ 9 + 1
-6 ≤ x ≤ 10
Interval Notation:
[-6, 10]
Graphing the Interval: Example 5: Solve each inequality, write in interval notation, graph.
-90 < -9x ≤ -27
To isolate x in the middle, let's divide each part by (-9). Remember this means we have to flip each inequality symbol.
-90/-9 > -9/-9 x ≥ -27/-9
10 > x ≥ 3
Write this in the direction of the number line:
3 ≤ x < 10
Interval Notation:
[3, 10)
Graphing the Interval:

---

Skills Check: