Problem 60 - College Algebra

Question

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$|x+3| \leq 4$$

Step-by-Step Solution

Answer

The solution to the inequality \(|x+3| \leq 4\) is \([-7, 1]\) in interval notation.

Step 1: Rewrite the Absolute Value Inequality as a Compound Inequality

An absolute value of an expression being less than or equal to a number can be written as a compound inequality where the expression is both greater than or equal to the negative of the number and less than or equal to the number. Therefore, the inequality \(|x+3| \leq 4\) can be rewritten as \(-4 \leq x+3 \leq 4\).

Step 2: Solve the Compound Inequality for x

Next, solve the compound inequality \(-4 \leq x+3 \leq 4\) for x by subtracting 3 from all parts of the inequality. This yields \(-4-3 \leq x+3-3 \leq 4-3\), which simplifies to \(-7 \leq x \leq 1\). So all numbers x that satisfy the inequality \(|x+3| \leq 4\) are the numbers between -7 and 1, inclusive.

Step 3: Draw the Solution Set on a Number Line

To represent this solution set on a number line, draw a line with numbers marked from -7 to 1. Since the inequality includes both -7 and 1, make a solid dot at -7 and 1 to indicate that these values are included in the solution set.

Step 4: Express the Solution Set in Interval Notation

Lastly, to express the solution set in interval notation, write the solution set as the interval \([-7, 1]\). This means that the solution set consists of all numbers between -7 and 1, inclusive.

Key Concepts

Compound Inequalities

When dealing with absolute value inequalities like \(|x+3| \leq 4\), it's helpful to rewrite them as compound inequalities. Absolute value expressions can be tricky, but this method makes them more manageable. When an absolute value is less than a number, it implies the expression inside the absolute value is within a certain range. Here, we break it down into two separate inequalities:

• The expression is greater than or equal to the negative of the number.
• The expression is less than or equal to the number.

This turns \(|x+3| \leq 4\) into the compound inequality \(-4 \leq x+3 \leq 4\). This means the expression \(x+3\) lies between -4 and 4. By solving this inequality, we can find the range of values for \(x\).

Interval Notation

Once a compound inequality is solved, it's time to express the solution using interval notation. This notation provides a concise way to represent the set of solutions to an inequality. For the inequality \-7 \leq x \leq 1\, interval notation is used to show that \(x\) can take any value from -7 to 1, inclusive.
The brackets in interval notation have specific meanings:

• [ ] include the endpoints, meaning the values at the boundaries are part of the solution.
• ( ) exclude the endpoints, meaning the boundary values are not part of the solution.

Thus, the solution \(-7 \leq x \leq 1\) is written as \([-7, 1]\) in interval notation. This informs us that both -7 and 1 are included in the solution set.

Graphing Solution Sets

Graphing inequality solutions offers a visual aid to understanding. For \(-7 \leq x \leq 1\), we represent this on a number line to visually express the range of solutions. Start by drawing a horizontal line and mark critical points such as -7 and 1.
Here's how to graph this specific problem:

• Mark a solid line or shading between -7 and 1.
• Place solid dots at -7 and 1 to show these endpoints are included.

The result is a clear visual representation of all possible values for \(x\), showing that any number between -7 and 1, including the endpoints, is valid.

Number Line Representation

A number line provides an intuitive tool to display the solution to an inequality. For \(-7 \leq x \leq 1\), a number line helps clarify the range of viable numbers. Here's how to represent it:

• Draw a line with numbers across relevant points, especially focusing on -7 and 1.
• Create solid dots at -7 and 1 to indicate inclusiveness.
• Shade the line between these two points to show the continuous range of solutions.

This depiction aids in quickly understanding which values satisfy the inequality. It effectively communicates the set of solutions, bridging the gap between abstract mathematical concepts and visual comprehension.