Problem 61 - Algebra for College Students

Question

Solve each equation or inequality. Graph the solution set. $$ |3 x-1| \leq 11 $$

Step-by-Step Solution

Answer

The solution is \-\frac{10}{3} \leq x \leq 4\. Graph it on a number line from approximately -3.33 to 4, inclusive.

Step 1: Rewrite the Absolute Value Inequality

The absolute value inequality \(|3x-1| \leq 11\) can be rewritten as two linear inequalities: \(-11 \leq 3x-1 \leq 11\).

Step 2: Isolate the Middle Expression

Add 1 to all parts of the inequality to isolate the term with the variable inside the inequality: \(-11 + 1 \leq 3x - 1 + 1 \leq 11 + 1\). This simplifies to \-10 \leq 3x \leq 12\.

Step 3: Solve for x

Divide all parts of the compound inequality by 3 to isolate x: \(-10/3 \leq x \leq 12/3\). This simplifies to \-\frac{10}{3} \leq x \leq 4\.

Step 4: Graph the Solution Set

On a number line, graph the interval \([-\frac{10}{3}, 4]\). This includes all points from \~-3.33 to 4 inclusive. Use a solid dot or bracket at \(-\frac{10}{3}\) and 4 to indicate that these points are included in the solution set.

Key Concepts

absolute value inequalities

Absolute value inequalities might look tricky, but once you understand the concept, they're quite simple! The absolute value of a number, represented by vertical bars |x|, refers to its distance from zero on the number line.
When dealing with inequalities involving absolute values, you'll often need to split them into two different cases to solve. For example, the inequality \(|3x-1| \leq 11\) implies that the quantity inside the bars can be within the range of -11 to 11.

• This means you'll produce two separate inequalities:
  \(-11 \leq 3x-1 \leq 11\)


• This is the first step in solving the absolute value inequality by rewriting it in its compound inequality form.

compound inequalities

Compound inequalities involve two distinct inequalities that are connected by the words 'and' or 'or'. When we solved the absolute value inequality \(|3x-1| \leq 11\), it translated into the compound inequality \(-11 \leq 3x-1 \leq 11\).
This tells us the value of \(3x-1\) falls between -11 and 11.
To isolate x, follow these steps:

• Add 1 to all parts: \(-11 + 1 \leq 3x - 1 + 1 \leq 11 + 1\),
  which simplifies to \(-10 \leq 3x \leq 12\).
• Divide by 3: \(-10/3 \leq x \leq 12/3\), simplifies to \(-\frac{10}{3} \leq x \leq 4\).

The result, which is \(-\frac{10}{3} \leq x \leq 4\), represents the range of x values that satisfy our original absolute value inequality.

graphing inequalities

Graphing inequalities involves representing the solution sets of these inequalities on the number line.
In our exercise, we found that \(-\frac{10}{3} \leq x \leq 4\). This inequality tells us that x can be any value from \(-\frac{10}{3}\) to 4, inclusive.

• Draw a number line.
• Mark \(-\frac{10}{3}\) (approximately -3.33) and 4.
• Since the inequality includes -\frac{10}{3} and 4, place solid dots or brackets at these points.

This shaded region between \(-\frac{10}{3}\) and 4 represents all the x values that satisfy the original inequality.

algebra 1

In Algebra 1, students often encounter absolute value and compound inequalities. Understanding the steps to simplify and solve these inequalities helps build a solid foundation in algebra.
Key concepts covered here include:

• Breaking down absolute value inequalities
• Turning them into compound inequalities
• Solving step-by-step to isolate the variable
• Graphing the results on a number line

Mastering these skills is essential as they are fundamental in more advanced math topics.
Practice makes perfect! The more problems you solve, the more confident you'll become in handling inequalities.