Problem 24 - Algebra for College Students

Question

Solve each equation and inequality. \(|3 x+1| \leq 13\)

Step-by-Step Solution

Answer

The solution is \(-\frac{14}{3} \leq x \leq 4\).

Step 1: Understanding Absolute Value Inequality

The inequality given is an absolute value inequality: \(|3x+1| \leq 13\). This inequality says that the expression inside the absolute value, \(3x+1\), can be at most 13 units away from zero on the number line.

Step 2: Set Up Two Inequalities

The definition of absolute value says that \(|a| \leq b\) can be split into two inequalities: \(a \leq b\) and \(a \geq -b\). Therefore, for \(|3x+1| \leq 13\), we have two inequalities: (1) \(3x + 1 \leq 13\) (2) \(3x + 1 \geq -13\).

Step 3: Solve the First Inequality

Let's solve the first inequality, \(3x + 1 \leq 13\):Subtract 1 from both sides: \(3x \leq 12\)Then, divide both sides by 3: \(x \leq 4\).

Step 4: Solve the Second Inequality

Now, solve the second inequality, \(3x + 1 \geq -13\):Subtract 1 from both sides: \(3x \geq -14\)Then, divide both sides by 3: \(x \geq -\frac{14}{3}\).

Step 5: Combine the Solutions

Combine the two obtained solutions to write the interval for \(x\). The solutions are: \(x \leq 4\) and \(x \geq -\frac{14}{3}\). Thus, the combined solution is:\[-\frac{14}{3} \leq x \leq 4\].

Key Concepts

Inequality Solutions

When dealing with inequalities, our main goal is to find all possible values of the variable that make the inequality true. In the case of absolute value inequalities, like \(|3x + 1| \leq 13\), we face two scenarios.

• The expression inside the absolute value is less than or equal to the positive version of the number on the other side of the inequality.
• The expression inside the absolute value is greater than or equal to the negative version of that number.

By setting up these two separate inequalities, we are "splitting" the absolute value inequality into two parts, allowing us to comprehensively capture the range of solutions:1. \(3x + 1 \leq 13\)2. \(3x + 1 \geq -13\)We then solve these inequalities separately to find where the variable values overlap, which gives us the solution set. This method shows the power of breaking down complex inequalities into simpler parts.

Interval Notation

Interval notation is a way of expressing a range of values that solve an inequality in a concise format. After solving the inequalities, we ended up with two conditions for \(x\):

• \(x \leq 4\)
• \(x \geq -\frac{14}{3}\)

Write these conditions together to express the full range of values for \(x\): \[-\frac{14}{3} \leq x \leq 4\].In interval notation, this range is shown as: \([-\frac{14}{3}, 4]\).Here:

• The brackets \([\ and \)] indicate that both \(-\frac{14}{3}\) and \(4\) are included in the solution set because the inequality is inclusive (indicated by \(\leq\)).

Interval notation provides a neat and clear way of representing solution sets on a number line and is pivotal for communicating the solution effectively.

Absolute Value Properties

Understanding the properties of absolute value is crucial when working with absolute value inequalities. The absolute value of a number refers to its distance from zero on the number line, regardless of direction. Therefore, \(|a|\) represents this non-negative distance.The key properties to remember are:

• \(|a| \geq 0\) for any real number \(a\).
• \(|a| = a\) if \(a \geq 0\), and \(|a| = -a\) if \(a < 0\).
• For any inequality of the form \(|a| \leq b\), if \(b\) is positive, this can be reinterpreted as two separate inequalities: \(-b \leq a \leq b\).

By using these properties, we can effectively solve absolute value inequalities by breaking them into manageable parts. This allows us to find all potential solutions that satisfy the original condition.