Problem 60 - Elementary and Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. \(|4-3 x| \leq 13\)

Step-by-Step Solution

Answer

The solution is \([-3, \frac{17}{3}]\) in interval notation.

Step 1: 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 \(4 - 3x \leq 13\).

Step 2: 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 \(x \leq \frac{17}{3}\).

Step 3: 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:\[x \geq \frac{9}{-3}\]\[x \geq -3\]

Step 4: 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 \(\frac{17}{3}\).

Step 5: Represent Solution in Interval Notation

The interval notation for the solution \(-3 \leq x \leq \frac{17}{3}\) is:\[-3, \frac{17}{3}\].

Step 6: 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.

Key Concepts

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:

• \(-13 \leq 4 - 3x\)
• \(4 - 3x \leq 13\)

Both need to be satisfied simultaneously, to respect the original inequality. Each inequality must be solved separately. This process involves basic algebraic steps:

• Isolate the term with the variable \(x\)
• Remember to flip the inequality sign when multiplying or dividing by a negative number

By solving these, we find the range of values for \(x\) that maintain the balance within the given limits.

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
\([-3, \frac{17}{3}]\).

• 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.

Interval notation offers a neat and mathematical shorthand to express continuous sets of solutions without lengthy explanations. This helps effectively communicate solution ranges in both written and graphed formats.

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.

The shaded area represents all real number solutions for \(x\). It’s crucial that the endpoints' circles are filled, indicating the containment of those values. Graphing not only helps in checking solutions' correctness but also offers pictorial insight into the problem's nature.