Problem 59 - Elementary and Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. \(|3 x-2|<10\)

Step-by-Step Solution

Answer

The solution is \\(-\frac{8}{3} < x < 4\\) or \\((-\frac{8}{3}, 4)\\) in interval notation.

Step 1: Understanding the Absolute Inequality

First, we need to understand what the inequality \(|3x - 2| < 10\) means. It signifies that the distance of \(3x - 2\) from zero is less than 10. Thus, it results in two separate inequalities being true: \(3x - 2 < 10\) and \(-(3x - 2) < 10\). We will solve these inequalities separately.

Step 2: Solving the First Inequality

Let's solve the easy part first: \(3x - 2 < 10\).1. Add 2 to both sides: \3x < 12\2. Divide by 3: \x < 4\This inequality shows that \(x\) must be less than 4.

Step 3: Solving the Second Inequality

Now, let's solve the second inequality which is \(-(3x - 2) < 10\). Simplify this:1. Rewrite the expression: \-3x + 2 < 10\2. Subtract 2 from both sides: \-3x < 8\3. Divide by -3, and remember to reverse inequality sign: \x > -\frac{8}{3}\This tells us that \(x\) must be greater than \(-\frac{8}{3}\).

Step 4: Finding the Solution Set

Now we combine the two inequalities: \(-\frac{8}{3} < x < 4\).This represents the range of values \(x\) can take, satisfying the original absolute inequality.

Step 5: Graphing the Solution Set

Draw a number line:1. Locate \(-\frac{8}{3}\) and 4 on the line.2. Draw open circles at these points, since neither \(-\frac{8}{3}\) nor 4 are included in the solution set.3. Shade the region between \(-\frac{8}{3}\) and 4 to indicate all numbers in this interval are solutions.

Step 6: Writing the Solution in Interval Notation

The interval notation for the solution set is: \((-\frac{8}{3}, 4)\).This concisely represents all values of \(x\) that satisfy \(|3x - 2| < 10\).

Key Concepts

Absolute Value

An absolute value represents the distance a number is from zero on the number line. It transforms any negative value into a positive by considering only the magnitude, not the direction. In inequalities, such as \(|3x - 2| < 10\), the absolute value tells us that the expression \(3x - 2\) must always be less than 10 units away from zero.
To solve an inequality involving an absolute value, like \(|3x - 2| < 10\), it is split into two parts: one for the positive direction and one for the negative direction. For this particular problem:

• \(3x - 2 < 10\) handles the positive condition, meaning the expression is less than 10 naturally.
• \(-(3x - 2) < 10\) or equivalently \-3x + 2 < 10\ handles the negative scenario, reversing the expression.

These inequalities ensure that the absolute value rule is respected, and both must be solved to find all possible solutions that satisfy \(|3x - 2| < 10\).

Interval Notation

Interval notation provides a concise way to express a set of numbers that fall within certain boundaries. This is especially useful for expressing ranges in the context of inequalities. For the inequality \(-\frac{8}{3} < x < 4\), interval notation helps simplify the explanation of the solution set.
The parentheses in interval notation, like \((a, b)\), indicate that the endpoints \(a\) and \(b\) are not included in the set. If they were included, brackets \[a, b\] would be used.
In the solution \((-rac{8}{3}, 4)\), it means all the numbers greater than \(-\frac{8}{3}\) and less than \(4\) satisfy the inequality. Using interval notation helps avoid lengthy explanations about which values are included or excluded, and it is a very standard format in mathematics to express solutions succinctly.

Number Line Graphing

Graphing a solution on a number line is an intuitive way to visualize which values satisfy an inequality. In the problem \(-\frac{8}{3} < x < 4\), graphing it aids our understanding by making the range of solutions clear.
To create the number line graph for this example:

• First, identify and plot the endpoints \(-\frac{8}{3}\) and \(4\) on the number line.
• Draw open circles at these points to indicate they are not part of the solution set.
• Shade the region between \(-\frac{8}{3}\) and \(4\) to show that all numbers in this interval satisfy the inequality.

This visual approach makes it easy to see at a glance which numbers work for the inequality, reinforcing the solution set expressed in interval notation. It provides an immediate understanding of where the solution lies on the number line.