2.14: Algebraic Solutions to Absolute Value Inequalities
A ball is fired from the cannon during the Independence Day celebrations. It is fired directly into the air with an initial velocity of 150 ft/sec. The speed of the cannon ball can be calculated using the formula
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Khan Academy Absolute Value Inequalities
Guidance
You have learned that a linear inequality is of the form
Recall that when solving absolute value linear equations, you have to solve for the two related equations. Remember that for
The table below shows the four types of absolute value linear inequalities and the two related inequality expressions required to be solved for each one.
Absolute Value Inequality 





Equation 1 




Equation 2 




Remember the rules to algebraically solve for the variable remain the same as you have used before. It is also important to remember, as you found with absolute value linear equations, that if
Example A
Solve for the absolute value inequality
Solution:
Solutions
Example B
Solve for the absolute value inequality
Solution:
Solutions
Example C
Solve for the absolute value inequality
Solution:
Solutions
Concept Problem Revisited
A ball is fired from the cannon during the Independence Day celebrations. It is fired directly into the air with an initial velocity of 150 ft/sec. The speed of the cannon ball can be calculated using the formula
Therefore when
Vocabulary
 Absolute Value Linear Inequality

Absolute Value Linear inequalities can have one of four forms:
ax+b>c,ax+b<c,ax+b≥c , orax+b≤c . Absolute value linear inequalities have two related inequalities. For example forax+b>c , the two related inequalities areax+b>c andax+b>−c .
 Linear Inequality

Linear inequalities can have one of four forms:
ax+b>c,ax+b<c,ax+b≥c , orax+b≤c . In other words, the left side no longer equals the right side, it is less than, greater than, less than or equal to, or greater than or equal to.
Guided Practice
1. Solve for the solution set to the inequality \begin{align*}x1 \ge 9\end{align*}.
2. Solve for the solution set to the inequality \begin{align*}2w+7<23\end{align*}.
3. Solve for the solution set to the inequality \begin{align*}4+2b+3 \le 21\end{align*}.
Answers:
1. \begin{align*}x1 \ge 9\end{align*}
\begin{align*}x1 & \ge 9\\ x1{\color{red}+1} & \ge 9{\color{red}+1} && (\text{Add 1 to both sides to isolate and solve for the variable})\\ x & \ge 10\\ & OR\\ x1 & \le 9\\ x1{\color{red}+1} & \le 9{\color{red}+1} && (\text{Add 1 to both sides to isolate and solve for the variable})\\ x & \le 8\end{align*}
Therefore solution set is \begin{align*}x \ge 10,x \le 8\end{align*}.
2. \begin{align*}2w+7 < 23\end{align*}
\begin{align*}2w+7 &< 23\\ 2w+7{\color{red}7} &< 23{\color{red}7} && (\text{Subtract 7 from both sides to get variables on same side})\\ 2w &< 16 && (\text{Simplify})\\ \frac{2w}{{\color{red}2}} &> \frac{16}{{\color{red}2}} &&(\text{Divide by 2 to solve for the variable, reverse sign of inequality})\\ w &> 8\\ & OR \\ 2w+7 &> 23\\ 2w+7{\color{red}7} &> 23{\color{red}7} && ( \text{Subtract 7 from both sides to get variables on same side})\\ 2w &> 30 && (\text{Simplify})\\ \frac{2w}{{\color{red}2}} &< \frac{30}{{\color{red}2}} &&(\text{Divide by 2 to solve for the variable, reverse sign of inequality})\\ w &<15\end{align*}
Therefore solution set is \begin{align*}w >8,w<15\end{align*}.
3. \begin{align*}4+2b+3 \le 21\end{align*}
\begin{align*}4+2b+3 &\le 21\\ 2b1 & \le 21 && (\text{Combine constant terms on left side of inequality sign})\\ 2b1{\color{red}+1} & \le 21{\color{red}+1} && (\text{Add 1 to both sides to isolate the variable})\\ 2b & \le 22 && (\text{Simplify})\\ \frac{2b}{{\color{red}2}} &\le \frac{22}{{\color{red}2}} && (\text{Divide both sides by 2 to solve for the variable})\\ b & \le 11\\ & OR\\ 4+2b+3 & \ge 21\\ 2b1 & \ge 21 && (\text{Combine constant terms on left side of inequality sign})\\ 2b1{\color{red}+1} & \ge 21{\color{red}+1} && (\text{Add 1 to both sides to isolate the variable})\\ 2b & \ge 20 && (\text{Simplify})\\ \frac{2b}{{\color{red}2}} &\ge \frac{20}{{\color{red}2}} && (\text{Divide both sides by 2 to solve for the variable})\\ b & \ge 10\end{align*}
Therefore solution set is \begin{align*}b \le 11,b \ge 10\end{align*}.
Practice
Find the solution sets for the variable in each of the following absolute value linear inequalities.
 \begin{align*}p16>10\end{align*}
 \begin{align*}r+2<5\end{align*}
 \begin{align*}32k\ge 1\end{align*}
 \begin{align*}8y>5\end{align*}
 \begin{align*}8 \ge 5d2\end{align*}
Find the solution sets for the variable in each of the following absolute value linear inequalities.
 \begin{align*}s+25>8\end{align*}
 \begin{align*}10+8w2<16\end{align*}
 \begin{align*}2q+15 \le 7\end{align*}
 \begin{align*}\big \frac{1}{3}(g2) \big <4\end{align*}
 \begin{align*}2(e+4)>17\end{align*}
Find the solution sets for the variable in each of the following absolute value linear inequalities.
 \begin{align*}5x3(2x1)>3\end{align*}
 \begin{align*}2(a1.2)\ge 5.6\end{align*}
 \begin{align*}2(r+3.1) \le 1.4\end{align*}
 \begin{align*}\big\frac{3}{4}(m3)\big \le 8\end{align*}
 \begin{align*}\big2\left(e\frac{3}{4}\right)\big \ge 3\end{align*}
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Description
Learning Objectives
Here you'll learn how to solve an absolute value inequality.
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Date Created:
Dec 19, 2012Last Modified:
Apr 29, 2014Vocabulary
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