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practice and problem-solving ctice solve each equation. graph and check your solutions. 9. |b| = \frac{1}{2} 10. 4 = |y| 11. |n| + 3 = 7 12. 7 = |s| - 3 13. |x| - 10 = -2 14. 5|d| = 20 15. -3|m| = -9 16. |y| + 3 = 3 see problem 1. see problems 2 and 3.
To solve these absolute value equations, we'll use the definition of absolute value: if \(|x| = a\) (where \(a\geq0\)), then \(x = a\) or \(x = -a\). Let's solve each equation one by one.
Equation 9: \(|b|=\frac{1}{2}\)
Step 1: Apply the absolute value definition
By the definition of absolute value, if \(|b|=\frac{1}{2}\), then \(b=\frac{1}{2}\) or \(b = -\frac{1}{2}\).
Equation 10: \(4 = |y|\)
Step 1: Rewrite and apply the definition
We can rewrite this as \(|y| = 4\). By the absolute value definition, \(y = 4\) or \(y=-4\).
Equation 11: \(|n|+ 3=7\)
Step 1: Isolate the absolute value term
Subtract 3 from both sides of the equation:
\(|n|=7 - 3\)
\(|n| = 4\)
Step 2: Apply the absolute value definition
By the definition of absolute value, \(n = 4\) or \(n=-4\).
Equation 12: \(7=|s|-3\)
Step 1: Isolate the absolute value term
Add 3 to both sides of the equation:
\(|s|=7 + 3\)
\(|s|=10\)
Step 2: Apply the absolute value definition
By the definition of absolute value, \(s = 10\) or \(s=-10\).
Equation 13: \(|x|-10=-2\)
Step 1: Isolate the absolute value term
Add 10 to both sides of the equation:
\(|x|=-2 + 10\)
\(|x| = 8\)
Step 2: Apply the absolute value definition
By the definition of absolute value, \(x = 8\) or \(x=-8\).
Equation 14: \(5|d| = 20\)
Step 1: Isolate the absolute value term
Divide both sides of the equation by 5:
\(|d|=\frac{20}{5}\)
\(|d| = 4\)
Step 2: Apply the absolute value definition
By the definition of absolute value, \(d = 4\) or \(d=-4\).
Equation 15: \(-3|m|=-9\)
Step 1: Isolate the absolute value term
Divide both sides of the equation by \(- 3\):
\(|m|=\frac{-9}{-3}\)
\(|m| = 3\)
Step 2: Apply the absolute value definition
By the definition of absolute value, \(m = 3\) or \(m=-3\).
Equation 16: \(|y|+3 = 3\)
Step 1: Isolate the absolute value term
Subtract 3 from both sides of the equation:
\(|y|=3 - 3\)
\(|y| = 0\)
Step 2: Apply the absolute value definition
The only number whose absolute value is 0 is 0 itself. So \(y = 0\).
Final Answers:
- Equation 9: \(b=\frac{1}{2}\) or \(b = -\frac{1}{2}\)
- Equation 10: \(y = 4\) or \(y=-4\)
- Equation 11: \(n = 4\) or \(n=-4\)
- Equation 12: \(s = 10\) or \(s=-10\)
- Equation 13: \(x = 8\) or \(x=-8\)
- Equation 14: \(d = 4\) or \(d=-4\)
- Equation 15: \(m = 3\) or \(m=-3\)
- Equation 16: \(y = 0\)
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To solve these absolute value equations, we'll use the definition of absolute value: if \(|x| = a\) (where \(a\geq0\)), then \(x = a\) or \(x = -a\). Let's solve each equation one by one.
Equation 9: \(|b|=\frac{1}{2}\)
Step 1: Apply the absolute value definition
By the definition of absolute value, if \(|b|=\frac{1}{2}\), then \(b=\frac{1}{2}\) or \(b = -\frac{1}{2}\).
Equation 10: \(4 = |y|\)
Step 1: Rewrite and apply the definition
We can rewrite this as \(|y| = 4\). By the absolute value definition, \(y = 4\) or \(y=-4\).
Equation 11: \(|n|+ 3=7\)
Step 1: Isolate the absolute value term
Subtract 3 from both sides of the equation:
\(|n|=7 - 3\)
\(|n| = 4\)
Step 2: Apply the absolute value definition
By the definition of absolute value, \(n = 4\) or \(n=-4\).
Equation 12: \(7=|s|-3\)
Step 1: Isolate the absolute value term
Add 3 to both sides of the equation:
\(|s|=7 + 3\)
\(|s|=10\)
Step 2: Apply the absolute value definition
By the definition of absolute value, \(s = 10\) or \(s=-10\).
Equation 13: \(|x|-10=-2\)
Step 1: Isolate the absolute value term
Add 10 to both sides of the equation:
\(|x|=-2 + 10\)
\(|x| = 8\)
Step 2: Apply the absolute value definition
By the definition of absolute value, \(x = 8\) or \(x=-8\).
Equation 14: \(5|d| = 20\)
Step 1: Isolate the absolute value term
Divide both sides of the equation by 5:
\(|d|=\frac{20}{5}\)
\(|d| = 4\)
Step 2: Apply the absolute value definition
By the definition of absolute value, \(d = 4\) or \(d=-4\).
Equation 15: \(-3|m|=-9\)
Step 1: Isolate the absolute value term
Divide both sides of the equation by \(- 3\):
\(|m|=\frac{-9}{-3}\)
\(|m| = 3\)
Step 2: Apply the absolute value definition
By the definition of absolute value, \(m = 3\) or \(m=-3\).
Equation 16: \(|y|+3 = 3\)
Step 1: Isolate the absolute value term
Subtract 3 from both sides of the equation:
\(|y|=3 - 3\)
\(|y| = 0\)
Step 2: Apply the absolute value definition
The only number whose absolute value is 0 is 0 itself. So \(y = 0\).
Final Answers:
- Equation 9: \(b=\frac{1}{2}\) or \(b = -\frac{1}{2}\)
- Equation 10: \(y = 4\) or \(y=-4\)
- Equation 11: \(n = 4\) or \(n=-4\)
- Equation 12: \(s = 10\) or \(s=-10\)
- Equation 13: \(x = 8\) or \(x=-8\)
- Equation 14: \(d = 4\) or \(d=-4\)
- Equation 15: \(m = 3\) or \(m=-3\)
- Equation 16: \(y = 0\)