Problem 94 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ \left|2-\frac{x}{2}\right|-1 \leq 1 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(0 \leq x \leq 8\).

Step 1: Reformat inequality

Firstly, rearrange the given inequality \(\left|2-\frac{x}{2}\right|-1 \leq 1\) to isolate the absolute value. By adding 1 to both sides, the inequality becomes \(\left|2-\frac{x}{2}\right| \leq 2\). This form of inequality means that the quantity inside the absolute value is either less than or equal to 2, or greater than or equal to -2.

Step 2: Solve the Positive case

We'll first consider the 'positive' case where the quantity is less than or equal to 2: \[2-\frac{x}{2} \leq 2.\] This inequality simplifies to: \[-\frac{x}{2} \leq 0.\] By multiplying through by -2 (remembering to switch the direction of the inequality), the solution for 'x' becomes \(x \geq 0\).

Step 3: Solve the Negative case

Now consider the 'negative' case, where \(2-\frac{x}{2}\) is greater than or equal to -2. This simplifies to: \[2- \frac{x}{2} \geq -2.\] This further simplifies to: \[-\frac{x}{2} \geq -4.\] When multiplying this by -2 the inequality becomes \(x \leq 8\).

Step 4: Combine All Solutions

The final step is to combine the solutions for the positive and negative cases. The solution for the inequality is \(0 \leq x \leq 8\). Here, the final answer is the intersection of the solutions from the positive and negative cases.

Key Concepts

Solving Inequalities

When it comes to solving inequalities, like the one in our exercise, the main goal is to find the range of values that satisfy the inequality condition.
Inequalities come in different forms, such as less than, greater than, less than or equal to, and greater than or equal to. Each form provides information about how two expressions relate to each other.

To solve an inequality, follow these general steps:

• Perform operations such as adding, subtracting, multiplying, or dividing to isolate the variable on one side of the inequality.
• Remember to flip the inequality sign whenever you multiply or divide by a negative number.
• After solving, express the solution set in a way that shows all possible values that make the inequality true.

Inequalities can be shown on a number line to visually represent the solution set. For example, the solution to our exercise was \(0 \leq x \leq 8 \), which means all values between and including 0 and 8 satisfy the inequality.
This tells us that in terms of real-world applications, as long as "x" stays within this range, it fits the condition set by the inequality.

Absolute Value

The absolute value of a number or expression refers to its distance from zero on the number line, regardless of which direction. It is always a non-negative value, written with vertical bars such as \(| x | \).

In an absolute value inequality, such as the one given in the exercise, \( \left|2-\frac{x}{2}\right|-1 \leq 1 \), the expression inside the absolute value can be positive, negative, or zero; but the absolute value itself is always considered in non-negative terms.

This is crucial when solving absolute value inequalities, as the equality is split into two scenarios:

• The expression inside the absolute value is equal to a positive value.
• The expression inside the absolute value is equal to a negative value (equivalent in absolute terms, but opposite in sign).

The solution involves solving both scenarios (often split into a 'positive' and 'negative' case), then merging their results into a final solution that satisfies the original inequality condition.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations combined in a meaningful way.
The expression \(2-\frac{x}{2}\) from the exercise is an example.

Understanding how to manipulate and simplify algebraic expressions is key to solving problems like inequalities and equations. In our exercise, manipulating \(-\frac{x}{2}\) involved distribution and operations that are fundamental to swapping inequality sides and signs.

Here are some key tips for handling algebraic expressions:

• Simplify the expressions by combining like terms.
• Use distribution to handle terms multiplied by a variable or within parentheses.
• Pay attention to the order of operations: parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right).

By mastering these skills, you can navigate through complex expressions efficiently, paving the way for solving higher-level algebraic problems.