About Solving a Three-Part Inequality:

When we work with three-part linear inequalities, we find that we have three algebraic expressions separated by two inequality symbols. To find a solution, we are looking to isolate the variable in the middle. In other words, we want to find a solution where x is between two numbers.


Test Objectives
  • Demonstrate the ability to solve a three-part inequality
  • Demonstrate the ability to write a solution in interval notation
  • Demonstrate the ability to graph an interval on the number line
Solving Three-Part Inequalities Practice Test:

#1:

Instructions: solve each, write in interval notation, graph.

$$a)\hspace{.2em}-28 ≤ 7x ≤ 63$$

$$b)\hspace{.2em}0 ≤ \frac{x}{3}< 2$$


#2:

Instructions: solve each, write in interval notation, graph.

$$a)\hspace{.2em}3 ≤ 3 + x ≤ 1$$

$$b)\hspace{.2em}-15 < 3 - 3x ≤ 0$$


#3:

Instructions: solve each, write in interval notation, graph.

$$a)\hspace{.2em}-14 ≤ 2x - 10 ≤ 10$$

$$b)\hspace{.2em}-46 < 4x - 10 < -6$$


#4:

Instructions: solve each, write in interval notation, graph.

$$a)\hspace{.2em}-22 < -2 - 4x < 18$$

$$b)\hspace{.2em}23 ≤ 6x + 5 ≤ 53$$


#5:

Instructions: solve each, write in interval notation, graph.

$$a)\hspace{.2em}64 ≤ -8x - 8 ≤ 72$$

$$b)\hspace{.2em}9 < 8 + x < 12$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}-4 ≤ x ≤ 9, [-4,9]$$

$$b)\hspace{.2em}0 ≤ x < 6, [0,6)$$


#2:

Solutions:

$$a)\hspace{.2em}No \hspace{.2em}Solution, \hspace{.2em}∅$$

$$b)\hspace{.2em}1 ≤ x < 6, [1,6)$$

[0,6) graphed

#3:

Solutions:

$$a)\hspace{.2em}-2 ≤ x ≤ 10, [-2, 10]$$

$$b)\hspace{.2em}-9 < x < 1, (-9, 1)$$


#4:

Solutions:

$$a)\hspace{.2em}-5 < x < 5, (-5,5)$$

$$b)\hspace{.2em}3 ≤ x ≤ 8, [3,8]$$


#5:

Solutions:

$$a)\hspace{.2em}-10 ≤ x ≤ -9, [-10,-9]$$

$$b)\hspace{.2em}1 < x < 4, (1,4)$$