Problem 57 - Elementary and Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. \(|x+9| \leq 12\)

Step-by-Step Solution

Answer

The solution set is \([-21, 3]\) in interval notation. Plot \(-21\) to \(3\) with closed circles on a number line.

Step 1: Understand Absolute Value Inequality

We are given an absolute value inequality \(|x+9| \leq 12\). This means the expression inside the absolute value, \(x+9\), can be between \(-12\) and \(12\).

Step 2: Break Down the Inequality

The absolute value inequality \(|x+9| \leq 12\) can be rewritten as two inequalities: \(-12 \leq x+9 \leq 12\).

Step 3: Solve the Compound Inequality

To solve \(-12 \leq x+9 \leq 12\), begin by subtracting 9 from all parts of the inequality: \(-12 - 9 \leq x+9 - 9 \leq 12 - 9\). This simplifies to \(-21 \leq x \leq 3\).

Step 4: Write the Solution in Interval Notation

The interval where \(x\) satisfies the inequality \(-21 \leq x \leq 3\) is written in interval notation as \([-21, 3]\).

Step 5: Graph the Solution Set

Plot the interval \([-21, 3]\) on a number line. Draw a solid line between \(-21\) and \(3\) with closed circles at both endpoints to show that these points are included in the solution set.

Key Concepts

Interval Notation

Interval notation is a way of expressing a range of values on a number line. It is a handy form of writing the solution set for inequalities. When using interval notation, we represent the beginning and ending numbers of the interval using brackets. Here’s how to understand each type of bracket used:

• Square brackets [ ]: Indicate that the endpoint numbers are included in the interval. This is also known as a closed interval. For example, \([-21, 3]\) means the interval includes both -21 and 3.
• Parentheses ( ): Show that the endpoint numbers are not included, called an open interval. For instance, \((a, b)\) would mean everything between but not including "a" and "b".
• For a combination, such as \([-21, 3)\), it includes -21 but not 3.

Using interval notation offers a succinct and clear way of summarizing ranges, especially in problems involving inequalities. When reading or writing interval notation, always pay careful attention to brackets to understand what numbers are included or excluded.

Compound Inequalities

Compound inequalities involve finding the solutions to two or more inequalities at the same time. These can be either "and" or "or" inequalities.

• **"And" Compound Inequalities**: Both conditions in the inequality must be true. For example, \(-12 \leq x+9 \leq 12\) breaks down into two conditions: \(-12 \leq x+9\) and \(x+9 \leq 12\). These types of inequalities often lead to a solution where the variable is within a certain range.
• **"Or" Compound Inequalities**: At least one condition must be true. Solutions to these inequalities can be spread out over two or more intervals.

In the given problem, we solve \( -12 \leq x+9 \leq 12 \) by performing operations on all parts of the inequality to isolate "x". Subtracting 9 results in \( -21 \leq x \leq 3 \), clearly indicating a range of values satisfying the original inequality.

Graphing Inequalities

Graphing inequalities helps visualize the solutions more concretely, making it easier to grasp the concept of solution sets. To graph an inequality on a number line:

• Identify the interval endpoints. In our exercise, the points are \(-21\) and \(3\).
• Draw a number line and mark these points.
• Use a solid dot (or closed circle) on \(-21\) and \(3\) to show that these points are included in the solution set. This is because the inequality \(-21 \leq x \leq 3\) includes the values at the endpoints.
• Draw a solid line between \(-21\) and \(3\), illustrating that all numbers in between are solutions.

Visual representation gives a clearer view of what numbers satisfy the inequality. Graphing is particularly useful when you want to present the solution in a format that is easy to interpret at a glance.