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Chapter 9: Problem 79

Let \(g(x)=|-1+3(x+1)| .\) Find all values of \(x\) for which \(g(x) \leq 5\)

Short Answer

Expert verified
The values of \(x\) that satisfy the given inequality are contained in the interval \([-\frac{1}{3}, 2]\)

Step by step solution

01

STEP 1: Define function and the inequality

The given function is \(g(x)=|-1+3(x+1)|\). We needs to find all \(x\) for which \(g(x) \leq 5\), or in other words: \(-1+3(x+1) \leq 5\). Now we will solve this inequality by considering two scenarios, based on the two cases that the definition of absolute value provides.
02

STEP 2: Solve for \(a \geq 0\) case

If \(-1+3(x+1) \geq 0\), then the inequality can be written as \(-1+3(x+1) \leq 5\). Simplifying this inequality we can obtain \(x \leq \frac{6}{3} = 2\)
03

STEP 3: Solve for \(a < 0\) case

If \(-1+3(x+1) < 0\), then by the definition of absolute value, the inequality can be written as \( -(-1+3(x+1)) \leq 5\), which in turn simplifies to \(1-3x-3 \leq 5\) or \(3x \geq -1\), and further simplifies to \(x \geq -\frac{1}{3}\)
04

STEP 4: Combine all solutions

The solution to the inequality is the intersection of the solutions obtained in steps 2 and 3. Looking at the real number line, \(x\) can take any value from \(-\frac{1}{3}\) to \(2\). So, the solution is: \(-\frac{1}{3} \leq x \leq 2\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Inequalities
Absolute value inequalities involve expressions inside absolute value symbols, like \(|a|\). These inequalities can be a bit tricky. The absolute value of a number is its distance from zero on the number line, regardless of direction. So, \(|a| \leq b\) means that the expression inside the absolute value, \(a\), is within \(-b\) and \(b\). This creates two inequalities:
  • \(a \leq b\)
  • \(a \geq -b\)
In our exercise, the absolute value expression is \(-1 + 3(x+1)\). To solve \(|-1 + 3(x+1)| \leq 5\), break it down into two cases based on whether the expression inside is positive or negative. This helps us cover all possibilities.
Solving Inequalities
To solve the inequality \(|-1 + 3(x+1)| \leq 5\), break it into two cases. Case 1: \(-1 + 3(x+1) \geq 0\)
Here, the absolute value is positive or zero, so just drop the absolute value and solve:
  • \(-1 + 3(x+1) \leq 5\)
  • Simplify: \(3x + 2 \leq 5\)
  • Result: \(x \leq 2\)
Case 2: \(-1 + 3(x+1) < 0\)
If the expression is negative, multiply by -1 and change the inequality direction:
  • \(-(-1 + 3(x+1)) \leq 5\)
  • Simplify: \(3x + 2 \geq -1\)
  • Result: \(x \geq -\frac{1}{3}\)
Combine these results to find the overlap of solutions: \(-\frac{1}{3} \leq x \leq 2\). This range represents all possible solutions.
Real Number Line
The real number line is a visual representation of all possible solutions. When solving inequalities, it's helpful to visualize where values fall.
  • In our example, we are dealing with \(-\frac{1}{3} \leq x \leq 2\).
  • This means we have a segment on the number line from \(-\frac{1}{3}\) to \(2\).
  • Every point within this interval, including \(-\frac{1}{3}\) and \(2\), satisfies the inequality.
It's a useful tool to ensure you've captured all solutions and provides a clear picture of where solutions lie relative to each other and zero.

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Most popular questions from this chapter

What does a dashed line mean in the graph of an inequality?

Sketch the graph of the solution set for the following system of inequalities: $$\left\\{\begin{array}{l}y \geq n x+b(n<0, b>0) \\\y \leq m x+b(m>0, b>0) \end{array}\right.$$

Describe what is meant by the intersection of two sets. Give an example.

Consider the sets \(A=\\{1,2,3,4\\}\) and \(B=\\{3,4,5,6,7\\}\) a. Write the set consisting of elements common to both \(\operatorname{set} A\) and \(\operatorname{set} B\) b. Write the set consisting of elements that are members of set \(A\) or of set \(B\) or of both sets.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The smallest real number in the solution set of \(2 x>6\) is 4.
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