Problem 133 - Contemporary Precalculus

Question

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x+2| \geq 3$$

Step-by-Step Solution

Answer

Answer: The solutions for the given inequality are \(x \leq -5\) or \(x \geq 1\).

Step 1: Understand the geometric approach

The geometric approach means that we will split the absolute value expression into two separate inequalities, one for the positive case and one for the negative case. Then, we will solve each inequality separately and combine the results.

Step 2: Split the absolute value expression into two cases

Given \(|x+2| \geq 3\), we can create two separate inequalities. For the positive case, let \(x+2\) be positive, so we have: $$(x+2) \geq 3$$ For the negative case, let \(x+2\) be negative, so we have: $$-(x+2) \geq 3$$

Step 3: Solve the positive case inequality

Solve the positive inequality \((x+2) \geq 3\): $$x \geq 3-2 \Rightarrow x \geq 1$$

Step 4: Solve the negative case inequality

Solve the negative inequality \(-(x+2) \geq 3\): $$-x-2 \geq 3 \Rightarrow -x \geq 5 \Rightarrow x \leq -5$$

Step 5: Combine the solutions

Now we have two cases: Case 1 from positive, \(x \geq 1\) and case 2 from negative, \(x \leq -5\). This means that \(x\) can be any value less than or equal to \(-5\) or greater than or equal to \(1\). This is the solution for the given inequality. $$x \leq -5 \, \text{or} \, x \geq 1$$

Key Concepts

Geometric Approach

The geometric approach is a visual method for understanding absolute value inequalities. Imagine absolute value as measuring distance from zero on a number line.
The inequality \(|x+2| \geq 3\) translates into finding numbers whose distance from \( -2 \) is at least 3 units.

• This is like saying you need to be either 3 steps to the right or 3 steps to the left of \(-2\).

This understanding allows us to separate the problem into two parts, reflecting both directions on the number line.

Solving Inequalities

Solving inequalities involves finding all values for a variable that satisfy a condition. When we split the absolute value inequality into two parts:

• Positive case: \((x + 2) \geq 3\)
• Negative case: \(-(x + 2) \geq 3\)

Solving these individually gives us different conditions for the solutions. Each case tells us a part of the story about where solutions to the original problem lie on the number line.

Absolute Value

The absolute value of a number is its distance from zero, regardless of direction on a number line. It’s always positive or zero.

• The expression \(|x+2|\) indicates how far \(x + 2\) is from zero.
• A statement like \(|x+2| \geq 3\) implies that \(x + 2\) is at least 3 units from zero.

This concept forms the basis for splitting the inequality into two separate inequalities, each representing one side of the original absolute value expression.

Inequality Solutions

Our solutions to absolute value inequalities result in a range or combination of values. For \(|x+2| \geq 3\):

• From the positive case, we have \(x \geq 1\)
• From the negative case, we get \(x \leq -5\)

Combine these solutions, we realize they represent two sets on the number line:

• Values greater than or equal to 1
• Values less than or equal to -5

This means our solution is \(x \leq -5\) or \(x \geq 1\), capturing how the graph of solutions extends in both directions away from dash not including dash the space between -5 and 1.