Problem 75 - Intermediate Algebra

Question

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. $$ |3 x+2|+1>15 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -\frac{16}{3}) \cup (4, \infty)\).

Step 1: Isolate the absolute value

Subtract 1 from both sides to get the absolute value by itself. \[|3x + 2| > 14\]

Step 2: Set up the inequality cases

For an inequality \(|A| > B\), we consider two cases: 1. \(3x + 2 > 14\)2. \(3x + 2 < -14\)

Step 3: Solve the first case

Solve \(3x + 2 > 14\) by subtracting 2 from both sides, resulting in:\[3x > 12\]Then divide by 3:\[x > 4\]

Step 4: Solve the second case

Solve \(3x + 2 < -14\) by subtracting 2 from both sides, resulting in:\[3x < -16\]Then divide by 3:\[x < -\frac{16}{3}\]

Step 5: Combine the solutions and write in interval notation

The solution set from both cases is the union:\[x < -\frac{16}{3}\] or \[x > 4\]In interval notation, this is:\((-\infty, -\frac{16}{3}) \cup (4, \infty)\)

Step 6: Graph the solution set

Draw a number line, plot open circles at \(-\frac{16}{3}\) and 4, and shade the regions \((-\infty, -\frac{16}{3})\) and \((4, \infty)\) to indicate where the solution applies.

Key Concepts

Interval Notation

When dealing with inequalities, it is crucial to express the solution set clearly and concisely. That's where interval notation comes in. Interval notation is a simple way to write all the numbers between two given numbers or to express the range of solutions.

In our example, the solution is split into two separate intervals:

• For the inequality \(x < -\frac{16}{3}\), this is represented as \((-9, -\frac{16}{3})\). This indicates all numbers less than \(-\frac{16}{3}\), stretching infinitely leftward.
• For the inequality \(x > 4\), this is noted as \((4, \infty)\). This encompasses all numbers greater than 4, extending infinitely to the right.

These intervals are united by the union symbol \(\cup\), which means both sets of numbers are part of the solution.

Open intervals use parentheses \(()\) to indicate that the endpoint is not included in the set, perfect for inequalities with "<" or ">". If endpoints were included, brackets \([]\) would be used.

Graphing Inequalities

Visualizing inequalities through graphing is a helpful way to understand the solution set. It aids in confirming the range of values that satisfy the inequality. Here's how you can graph the solution:

1. **Draw a Number Line:** Start by sketching a simple horizontal line, which will represent all possible values of \(x\).

2. **Identify and Mark Key Points:** Locate and mark the critical values from the solution on the number line. In our case, these values are \(-\frac{16}{3}\) and 4. Use an open circle to symbolize that these points are not part of the solution.

3. **Shade the Solution Regions:** - For \(x < -\frac{16}{3}\), shade the portion of the line extending from \(-\frac{16}{3}\) towards the left. - For \(x > 4\), shade the section running to the right from 4.

This graphing process visually represents the intervals with open circles and shaded regions, reinforcing the range of solutions covered by the inequality.

Inequality Solutions

Absolute value inequalities might seem a bit daunting initially, but they follow a logical process. An absolute value inequality like \(|A| > B\) tells us that the value of \(A\) is more than \(B\) away from zero on either side of the number line.

The key is to break the absolute value inequality into two "cases":

• **Positive Case:** Assuming the expression inside the absolute value is positive. Like solving \(3x + 2 > 14\) in our example.
• **Negative Case:** Considering the expression inside can also be the negative equivalent, as in \(3x + 2 < -14\).

When solved, these cases provide the boundaries for \(x\). These solutions are not overlapping, which leads us to use the union operation between them when combining.

Lastly, each solution case is resolved independently to maintain clarity and break down the problem into simpler steps. These solutions then help us write accurate interval notation and graphing for a comprehensive understanding of the inequality.