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Magazine Chapter 1: Problem 37
Solve the inequality, and write the solution set in interval notation. \(3|4-x|-2<16\)
Short Answer
Expert verified
The solution set is \((-2, 10)\).
Step by step solution
01
- Isolate the Absolute Value Expression
First, add 2 to both sides of the inequality: \[3|4-x| - 2 + 2 < 16 + 2\]This simplifies to: \[3|4-x| < 18\]
02
- Divide Both Sides by 3
Next, divide both sides by 3 to isolate the absolute value expression: \[\frac{3|4-x|}{3} < \frac{18}{3}\]This simplifies to: \[|4-x| < 6\]
03
- Remove the Absolute Value
The expression \(|4-x| < 6\) means \[-6 < 4 - x < 6\]
04
- Solve the Compound Inequality
To solve \(-6 < 4 - x < 6\), first subtract 4 from all parts of the inequality: \[-6 - 4 < 4 - x - 4 < 6 - 4\]This simplifies to: \[-10 < -x < 2\]
05
- Divide by -1 and Reverse the Inequality Signs
Divide the entire inequality by -1 and reverse the inequality signs: \[10 > x > -2\] This can be rewritten as: \[-2 < x < 10\]
06
- Write in Interval Notation
Rewrite the solution set in interval notation: \[(-2, 10)\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
inequality solving
Solving inequalities can be a tricky process, but with a step-by-step approach, it's manageable. Inequalities compare values and show if one is greater, less than, or equal to another. Instead of using the equal sign, inequalities use symbols like <, >, 鈮�, and 鈮�.
When solving an inequality, follow these key steps:
When solving an inequality, follow these key steps:
- Step 1: Simplify both sides of the inequality if needed.
- Step 2: Isolate the variable term on one side.
- Step 3: Perform the same operation on all parts of the inequality.
- Step 4: If you multiply or divide by a negative number, remember to reverse the inequality sign.
interval notation
Once you have solved an inequality, it's common to express the solution in interval notation. Interval notation is a concise way of writing subsets of the real number line. It uses parentheses and brackets to show which numbers are included in the interval.
Here鈥檚 a quick rundown:
To convert the solution from inequality \(-2 < x < 10\) to interval notation, we use parentheses because neither endpoint is included in the set.
Here鈥檚 a quick rundown:
- Parentheses ( , ): Use these to denote that the endpoints are not included in the interval.
- Brackets [ , ]: Use these when the endpoints are included.
To convert the solution from inequality \(-2 < x < 10\) to interval notation, we use parentheses because neither endpoint is included in the set.
absolute value
Absolute value represents the distance of a number from zero on the number line, regardless of direction. It鈥檚 always non-negative. This concept is crucial when solving equations or inequalities that include absolute value expressions.
Here are some key points:
Here are some key points:
- Definition: The absolute value of a number \( |a| \) is defined as \(a\) if \(a\geq0\), and \(-a\) if \(a<0\).
- Equations: To solve \( |a| = b \) where \(b > 0\), you need to consider both possibilities: \(a = b\) and \(a = -b\).
- Inequalities: For \( |a| < b \), you set up a compound inequality: \(-b < a < b\).
