Three-Part Inequality Lesson

In some cases, we will see what is known as a "three-part" inequality. To solve a three-part inequality, we 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 1: Solve each inequality, write in interval notation, graph.
-11 ≤ 3x - 5 ≤ -2
Since 5 is being subtracted away from x, we need to add 5 to each part:
-11 + 5 ≤ 3x - 5 + 5 ≤ -2 + 5
-6 ≤ 3x ≤ 3
We will divide each part by 3, the coefficient of x: $$\require{cancel}\frac{-6}{3}≤ \frac{3}{3}x ≤ \frac{3}{3}$$ $$\frac{-2\cancel{6}}{\cancel{3}}≤ \frac{1\cancel{3}}{\cancel{3}}x ≤ \frac{1\cancel{3}}{\cancel{3}}$$ $$-2 ≤ x ≤ 1$$ Interval Notation:
[-2, 1]
Graphing the Interval on the Number Line: Example 2: 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 3: 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:

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Skills Check: