Problem 63 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ |x+3| \geq 3 $$

Step-by-Step Solution

Answer

The solution is \(x \geq 0\) or \(x \leq -6\).

Step 1: Understand the Absolute Value Inequality

The inequality given is \(|x+3| \geq 3\). Absolute value inequalities of this form, \(|A| \geq B\), can be split into two separate inequalities: \(A \geq B\) or \(A \leq -B\). This is because the absolute value of a number is its distance from zero, which is always non-negative.

Step 2: Split the Inequality

Split the absolute value inequality \(|x+3| \geq 3\) into two separate inequalities: 1. \(x+3 \geq 3\)2. \(x+3 \leq -3\)

Step 3: Solve the First Inequality

Solve the inequality \(x+3 \geq 3\). Subtract 3 from both sides to isolate \(x\): \(x+3 - 3 \geq 3 - 3\) \(x \geq 0\).

Step 4: Solve the Second Inequality

Solve the inequality \(x+3 \leq -3\). Subtract 3 from both sides to isolate \(x\): \(x+3 - 3 \leq -3 - 3\) \(x \leq -6\).

Step 5: Combine the Solutions

Combine the solutions from both inequalities. The solution to \(x+3 \geq 3\) is \(x \geq 0\), and the solution to \(x+3 \leq -3\) is \(x \leq -6\). Thus, the solution set is \(x \geq 0\) or \(x \leq -6\).

Key Concepts

Absolute Value Inequalities

When dealing with absolute value inequalities, it helps to first understand the nature of absolute values. The absolute value of a number represents its distance from zero on a number line, and this distance is always positive or zero. In mathematical terms, if \(|A| \geq B\), the expression can represent two possible scenarios because it encompasses the entire range of numbers whose distance is greater than or equal to \(B\).

• First scenario: The number itself (\(A\)) could be greater than or equal to \(B\) if it's positive.
• Second scenario: The number could also be less than or equal to \(-B\) if it's negative.

For the inequality \(|x+3| \geq 3\), this translates to solving two separate inequalities:

• \(x+3 \geq 3\)
• \(x+3 \leq -3\)

This allows us to find the full range of solutions where the original absolute inequality holds true.

Inequality Solutions

Solving inequalities follows similar principles to solving equations, with special attention to the direction and nature of the inequality symbol. When you solve an absolute value inequality like \(|x+3| \geq 3\), each resulting inequality from the split needs to be solved separately. For the inequality \(x+3 \geq 3\):

• Subtract 3 from both sides to isolate \(x\).
• The inequality then becomes \(x \geq 0\).

For the inequality \(x+3 \leq -3\):

• Subtract 3 from both sides to isolate \(x\).
• This inequality converts to \(x \leq -6\).

The intersection or union of solutions offers a comprehensive view of the inequality resolution: here, it means the values satisfying the inequality are either \(x \geq 0\) or \(x \leq -6\). A correct solution must honor the logic of inequalities, ensuring that any algebraic steps maintain the proper inequality direction.

Algebraic Manipulation

Algebraic manipulation is essential for maintaining balance in both equations and inequalities. When working with inequalities, consider each manipulation step as carefully as you would in solving an equation. However, note one crucial difference: when multiplying or dividing by a negative number, the inequality sign must be flipped to preserve the mathematical truth.In our exercise, every manipulation involved keeping the inequality sign as is, because all operations on both sides involved positive numbers or like terms. Here’s how:

• For \(x+3 \geq 3\): Subtract 3 to obtain \(x \geq 0\).
• For \(x+3 \leq -3\): Subtract 3 to get \(x \leq -6\).

By understanding these manipulation rules, you can adeptly solve complex problems while preserving the integrity of the inequalities. Attention to detail in these steps ensures precise solutions and a deepened comprehension of algebraic strategies applied in solving inequalities.