About Solving Linear Inequalities:

Solving a linear inequality in one variable is similar to solving a linear equation in one variable. Our goal is still to isolate the variable on one side, with a number on the other side. We must always remember to flip the inequality symbol when multiplying or dividing by a negative number.


Test Objectives
  • Demonstrate the ability to use the addition property of inequality
  • Demonstrate the ability to use the multiplication property of inequality
  • Demonstrate the ability to solve a linear inequality in one variable
Solving Linear Inequalities Practice Test:

#1:

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

$$a)\hspace{.2em}315 ≥ -5(1 - 8x)$$

$$b)\hspace{.2em}-133 > 7(2x - 3)$$


#2:

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

$$a)\hspace{.2em}-106 < -2(7x - 3)$$

$$b)\hspace{.2em}\frac{309}{56}+ 6x ≥ -\frac{6}{7}x - \frac{1}{2}x$$


#3:

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

$$a)\hspace{.2em}-\frac{5}{3}x + \frac{3}{2}≤ -\frac{1}{3}- \frac{7}{2}x$$

$$b)\hspace{.2em}4(x + 5) - 5 ≥ -(3x - 3) + 5$$


#4:

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

$$a)\hspace{.2em}4(x - 1) ≤ -2(x + 5)$$

$$b)\hspace{.2em}5(x - 1) < 4(x - 4) + 2$$


#5:

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

$$a)\hspace{.2em}4(x + 1) - 2(x + 2) > -5x - 3x$$

$$b)\hspace{.2em}x - 4(3x + 3) ≤ 4(-3x - 3) + 4x$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}x ≤ 8, (-\infty, 8]$$

x is less than or equal to 8

$$b)\hspace{.2em}x < -8, (-\infty, -8)$$

x is less than -8

#2:

Solutions:

$$a)\hspace{.2em}x < 8, (-\infty, 8)$$

x is less than 8

$$b)\hspace{.2em}x ≥ -\frac{3}{4}, \left[-\frac{3}{4}, \infty\right)$$

x is greater than or equal to -3/4

#3:

Solutions:

$$a)\hspace{.2em}x ≤ -1, (-\infty, -1]$$

x is less than or equal to -1

$$b)\hspace{.2em}x ≥ -1, [-1, \infty)$$

x is greater than or equal to -1

#4:

Solutions:

$$a)\hspace{.2em}x ≤ -1, (-\infty, -1]$$

x is less than or equal to -1

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

x is less than -9

#5:

Solutions:

$$a)\hspace{.2em}x > 0, (0, \infty)$$

x is greater than 0

$$b)\hspace{.2em}x ≥ 0, [0, \infty)$$

x is greater than or equal to 0