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 begin by simplifying each side, which means clearing any fractions, decimals, or parentheses, and then combining like terms. Afterwards, we isolate the variable term on one side of the inequality and place a single number on the other side. Finally, we isolate the variable to obtain our solution. Remember, we must always flip the direction of 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 multi-step linear inequality in one variable
  • Demonstrate the ability to solve a linear inequality with parentheses
  • Demonstrate the ability to solve a linear inequality with decimals
  • Demonstrate the ability to solve a linear inequality with fractions
Solving Linear Inequalities Practice Test:

#1:

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

$$a)\hspace{.2em}315 ≥ -5(1 - 8x)$$

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

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#2:

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

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

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

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#3:

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

$$a)\hspace{.2em}\frac{367}{168}+ x + \frac{1}{3}+ \frac{7}{8}x ≥ \frac{1}{3}x + \frac{13}{7}$$

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

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#4:

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

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

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

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#5:

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

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

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

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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

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#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

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#3:

Solutions:

$$a)\hspace{.2em}x ≥ -\frac{3}{7}, \left[-\frac{3}{7}, \infty\right)$$x is greater than or equal to -3/7

$$b)\hspace{.2em}x ≥ -1, [-1, \infty)$$x is greater than or equal to -1

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#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

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#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

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