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Magazine Chapter 8: Problem 60
Solve each inequality. Graph the solution set and write it using interval notation. \(|4-3 x| \leq 13\)
Short Answer
Expert verified
The solution is \([-3, \frac{17}{3}]\) in interval notation.
Step by step solution
01
Understanding the Absolute Value Inequality
The inequality \(|4-3x| \leq 13\) involves an absolute value. An absolute value inequality \(|A| \leq B\) means that \(-B \leq A \leq B\). Here, \(A\) is \(4 - 3x\) and \(B\) is \(13\). Therefore, our inequality becomes two separate inequalities: \(-13 \leq 4 - 3x\) and \(4 - 3x \leq 13\).
02
Solve the First Inequality
Solve \(-13 \leq 4 - 3x\). First, subtract \(4\) from both sides to isolate the term involving \(x\):\[-13 - 4 \leq -3x\]\[-17 \leq -3x\]Now, divide both sides by \(-3\), remembering to flip the inequality sign because we are dividing by a negative number:\[\frac{-17}{-3} \geq x\]\[\frac{17}{3} \geq x\] This simplifies to \(x \leq \frac{17}{3}\).
03
Solve the Second Inequality
Now solve \(4 - 3x \leq 13\). Similarly, subtract \(4\) from both sides:\[4 - 3x - 4 \leq 13 - 4\]\[-3x \leq 9\]Then divide by \(-3\) and flip the inequality sign:\[x \geq \frac{9}{-3}\]\[x \geq -3\]
04
Combine the Inequalities
Combine the results of both inequalities to get the solution:\[-3 \leq x \leq \frac{17}{3}\]This means the solution for \(|4 - 3x| \leq 13\) in terms of \(x\) is \(x\) values including and between \(-3\) and \(\frac{17}{3}\).
05
Represent Solution in Interval Notation
The interval notation for the solution \(-3 \leq x \leq \frac{17}{3}\) is:\[-3, \frac{17}{3}\].
06
Graph the Solution Set
To graph the solution, draw a number line and shade the region between \(-3\) and \(\frac{17}{3}\), including those points. This represents all \(x\) values satisfying the inequality.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value Inequality
An absolute value measures how far a number is from zero, regardless of direction. When dealing with an absolute value inequality like \(|4 - 3x| \leq 13\), this means the expression inside the absolute value, \(4 - 3x\), must be within the range \(-13\) to \(13\).
This concept breaks into two inequalities:
This concept breaks into two inequalities:
- \(-13 \leq 4 - 3x\)
- \(4 - 3x \leq 13\)
- Isolate the term with the variable \(x\)
- Remember to flip the inequality sign when multiplying or dividing by a negative number
Interval Notation
Interval notation offers a streamlined way to describe a set of numbers. After solving inequalities, the solution set often includes all numbers between two endpoints. For example, in the inequality \(-3 \leq x \leq \frac{17}{3}\), interval notation is written as
\([-3, \frac{17}{3}]\).
\([-3, \frac{17}{3}]\).
- Brackets [ ] indicate that the endpoints are included, signifying there's equality at the boundary.
- The interval \([-3, \frac{17}{3}]\) includes every number between \(-3\) and \(\frac{17}{3}\), precisely showing the solution set compactly and clearly.
Graphing Inequalities
Graphing inequalities involves visually displaying the solution on a number line. This helps you see, at a glance, which numbers fulfill the inequality. Once the inequality is solved and expressed in interval notation, the next step is plotting:
- Draw a number line and mark key points, such as endpoints and important decimal fractions.
- Shade the region between \(-3\) and \(\frac{17}{3}\), as defined by \([-3, \frac{17}{3}]\) in interval notation.
- Include filled circles at \(-3\) and \(\frac{17}{3}\) to show these bounds are part of the solution set.
