Problem 12 - Algebra for College Students

Question

Solve each inequality and graph the solutions. \(|x+1|>3\)

Step-by-Step Solution

Answer

The solution is \((-\infty, -4) \cup (2, \infty)\).

Step 1: Understanding the Absolute Value Inequality

The inequality \(|x+1| > 3\) involves an absolute value, which represents the distance from zero. Solving \(|x + 1| > 3\) means finding the values of \(x\) that are more than 3 units from -1.

Step 2: Breaking Down the Inequality

The expression \(|x + 1| > 3\) means there are two cases to consider: \(x + 1 > 3\) and \(x + 1 < -3\), because the distance can be more than 3 in both positive and negative directions.

Step 3: Solving the First Case

Let's solve the inequality \(x + 1 > 3\). Subtract 1 from both sides to get \(x > 2\). This represents the values of \(x\) for which \(x + 1\) is more than 3.

Step 4: Solving the Second Case

Now solve the inequality \(x + 1 < -3\). Subtract 1 from both sides to get \(x < -4\). This represents the values of \(x\) for which \(x + 1\) is less than -3.

Step 5: Combining the Solutions

Combine the two inequalities \(x > 2\) and \(x < -4\). The solution to the inequality \(|x + 1| > 3\) is two separate intervals: \((-fty, -4)\) and \((2, fty)\).

Step 6: Graphing the Solutions

To graph the solutions, draw two separate parts on a number line: one interval is from \(-fty\) to \(-4\), and the other is from \(2\) to \(fty\). Use open circles at \(-4\) and \(2\) to indicate that these points are not included in the solution set.

Key Concepts

Solving Inequalities

Inequalities involve finding the range of values that satisfy certain conditions. When dealing with absolute value inequalities like \(|x + 1| > 3\), we are looking at the distance of two points on the number line. Absolute value represents the magnitude of a number, regardless of its sign.
To solve the inequality \(|x + 1| > 3\), we need to split it into two parts. This is because the absolute value can represent two different scenarios:

• The expression inside the absolute value is greater than 3, i.e., \(x + 1 > 3\).
• The expression inside the absolute value is less than -3, i.e., \(x + 1 < -3\).

These cases cover both the positive and negative distances from zero. Solving each scenario separately, we get:

• From \(x + 1 > 3\), we subtract 1 to find \(x > 2\).
• From \(x + 1 < -3\), we subtract 1 to find \(x < -4\).

These results tell us that the solution for the inequality are those values of \(x\) that are greater than 2 or less than -4.

Graphing Solutions

Graphing the solutions of inequalities can provide a visual representation of the solution set. This helps in understanding where the solutions lie on a number line.
For the inequality \(|x + 1| > 3\), the solution contains two parts: \(x > 2\) and \(x < -4\). These are illustrated on a number line by shading the regions corresponding to these inequalities.

• For \(x > 2\), draw a line extending to the right from the point 2. Use an open circle at 2 to indicate that 2 is not included in the solution.
• For \(x < -4\), draw a line extending to the left from the point -4. Again, use an open circle at -4 to show it's not part of the solution.

The open circles are used because the inequality is strict (\(>\) and \(<\), not \(\geq\) or \(\leq\)).
This visual representation clearly shows the solution intervals on the number line.

Mathematical Intervals

Mathematical intervals describe a range of numbers along the number line. In our example with the absolute value inequality \(|x + 1| > 3\), the solution consists of two separate intervals:

• The first interval is \((-\infty, -4)\), describing numbers less than -4.
• The second interval is \((2, \infty)\), consisting of numbers greater than 2.

These intervals are disjoint, meaning there's a gap between them.
Intervals are usually written in parentheses to show that the endpoints are not included (in the case of strict inequalities) or in brackets if they are included. Here, since the inequality is \(>3\), open intervals are used, with parentheses indicating that -4 and 2 aren't part of the solution set.
This method of expressing solutions is efficient and helpful in conveying the set of all possible answers in mathematical problems.