Solving Absolute Value Inequalities
Solving inequalities requires special techniques that you didn't need when you were solving equations.
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This guy does a great job of explaining how to solve inequalities containing absolute values.
video by MIP4U |
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Now try the practice problems.
Practice Solving Inequalities
Unless otherwise instructed, find all values of \(x\) that solve each inequality.
\(|x-2| < 4\)
Problem Statement |
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Find the values of \(x\) that solve the inequality \(|x-2| < 4\).
Final Answer |
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\(-2 < x < 6\)
Problem Statement
Find the values of \(x\) that solve the inequality \(|x-2| < 4\).
Solution
video by Dr Chris Tisdell |
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Final Answer
\(-2 < x < 6\)
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\(|3x-2| < 4\)
Problem Statement |
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Find the values of \(x\) that solve the inequality \(|3x-2| < 4\).
Final Answer |
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\(-2/3 < x < 2\)
Problem Statement
Find the values of \(x\) that solve the inequality \(|3x-2| < 4\).
Solution
video by Dr Chris Tisdell |
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Final Answer
\(-2/3 < x < 2\)
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\(|1-3x| \leq 2\)
Problem Statement |
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Find the values of \(x\) that solve the inequality \(|1-3x| \leq 2\).
Final Answer |
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\(-1/3 \leq x \leq 1\)
Problem Statement
Find the values of \(x\) that solve the inequality \(|1-3x| \leq 2\).
Solution
video by Dr Chris Tisdell |
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Final Answer
\(-1/3 \leq x \leq 1\)
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\(|5-4x| > 2\)
Problem Statement |
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Find the values of \(x\) that solve the inequality \(|5-4x| > 2\)
Final Answer |
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\(x < 3/4\) or \(x > 7/4\)
Problem Statement
Find the values of \(x\) that solve the inequality \(|5-4x| > 2\)
Solution
video by Dr Chris Tisdell |
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Final Answer
\(x < 3/4\) or \(x > 7/4\)
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\( 3|x+7| > 27 \)
Problem Statement
Find the values of \(x\) that solve the inequality \( 3|x+7| > 27 \).
Solution
video by Freshmen Math Doctor |
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\( |x+3| \leq 4 \)
Problem Statement
Find the values of \(x\) that solve the inequality \( |x+3| \leq 4 \).
Solution
video by Freshmen Math Doctor |
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\( |x| < 4 \)
Problem Statement |
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Find the values of \(x\) that solve \( |x| < 4 \).
Final Answer |
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\( -4 < x < 4 \)
Problem Statement
Find the values of \(x\) that solve \( |x| < 4 \).
Solution
Final Answer
\( -4 < x < 4 \)
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\( |2x - 3| \geq 8 \)
Problem Statement |
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Find the values of \(x\) that solve \( |2x - 3| \geq 8 \).
Final Answer |
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\( x \geq 5.5 \) or \( x \leq -2.5 \)
Problem Statement
Find the values of \(x\) that solve \( |2x - 3| \geq 8 \).
Solution
Final Answer
\( x \geq 5.5 \) or \( x \leq -2.5 \)
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\( 5 - 3|4x+1| \geq -9 \)
Problem Statement |
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Find the values of \(x\) that solve \( 5 - 3|4x+1| \geq -9 \).
Final Answer |
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\( -17/12 \leq x \leq 11/12 \)
Problem Statement
Find the values of \(x\) that solve \( 5 - 3|4x+1| \geq -9 \).
Solution
Final Answer
\( -17/12 \leq x \leq 11/12 \)
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\( |2x + 5| > 11 \)
Problem Statement |
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Find the values of \(x\) that solve \( |2x + 5| > 11 \)
Final Answer |
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\( x > 3 \) or \( x < -8 \)
Problem Statement
Find the values of \(x\) that solve \( |2x + 5| > 11 \)
Solution
Final Answer
\( x > 3 \) or \( x < -8 \)
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\( |3x - 4| \leq 17 \)
Problem Statement |
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Find the values of \(x\) that solve \( |3x - 4| \leq 17 \).
Final Answer |
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\( 13/3 \leq x \leq 7 \)
Problem Statement
Find the values of \(x\) that solve \( |3x - 4| \leq 17 \).
Solution
Final Answer
\( 13/3 \leq x \leq 7 \)
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\( |3x + 5| \leq -3 \)
Problem Statement |
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Find the values of \(x\) that solve \( |3x + 5| \leq -3 \)
Final Answer |
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no solution
Problem Statement
Find the values of \(x\) that solve \( |3x + 5| \leq -3 \)
Solution
Final Answer
no solution
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\( |4x - 3| > -4 \)
Problem Statement |
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Find the values of \(x\) that solve \( |4x - 3| > -4 \)
Final Answer |
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all real numbers
Problem Statement
Find the values of \(x\) that solve \( |4x - 3| > -4 \)
Solution
Final Answer
all real numbers
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\( 3|7x - 4| +5 > 17 \)
Problem Statement |
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Find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \)
Final Answer |
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\(x > 8/7 \) or \( x < 0 \)
Problem Statement
Find the values of \(x\) that solve \( 3|7x - 4| +5 > 17 \)
Solution
Final Answer
\(x > 8/7 \) or \( x < 0 \)
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\( 2 \leq |3x+5| \leq 10 \)
Problem Statement
Find the values of \(x\) that solve \( 2 \leq |3x+5| \leq 10 \) giving your answer in interval notation in exact form.
Solution
video by Michael Penn |
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\( ||3x-1|+4| \lt 8 \)
Problem Statement
Find the values of \(x\) that solve \( ||3x-1|+4| \lt 8 \) giving your answer in interval notation in exact form.
Solution
video by Michael Penn |
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