Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |x-4| \leq 0.03 $$

Step-by-Step Solution

Answer

The solution is \([3.97, 4.03]\).

Step 1: Understanding the Absolute Value Inequality

We are given the inequality \(|x-4| \leq 0.03\). This means that the distance between \(x\) and 4 on the number line must be less than or equal to 0.03. We will solve it using the definition of absolute value inequalities.

Step 2: Set the Inequality Equation

The inequality \(|x-4| \leq 0.03\) translates to two inequalities: \(x - 4 \leq 0.03\) and \(x - 4 \geq -0.03\).

Step 3: Solve Each Inequality

Solving the first inequality, \(x - 4 \leq 0.03\), we get:\[x \leq 4 + 0.03 = 4.03\]For the second inequality, \(x - 4 \geq -0.03\), we get:\[x \geq 4 - 0.03 = 3.97\]

Step 4: Determine the Interval

The solution to the inequality \(|x-4| \leq 0.03\) is the intersection of the solutions from the above two inequalities. It is given by:\[3.97 \leq x \leq 4.03\]

Step 5: Express the Solution as an Interval

The interval representation of the solution is:\([3.97, 4.03]\). This means \(x\) can be any number between 3.97 and 4.03, inclusive.

Key Concepts

Interval Notation

Interval notation is a powerful and concise way to express a range of numbers. It is particularly useful when dealing with solutions to inequalities, like in our exercise. Here’s an easy way to understand interval notation: If you have a range of numbers from 3.97 to 4.03, and you want to include both 3.97 and 4.03, you represent this as \([3.97, 4.03]\). The square brackets, \([\ ]\), indicate that the endpoints are included in the interval. Conversely, if we want to exclude endpoints, we use round brackets, or parenthesis \((\ )\).

• For inclusive intervals, use square brackets: \([a, b]\)
• For exclusive intervals, use round brackets: \((a, b)\)

In our solution, because the inequality \(|x - 4| \leq 0.03\) includes all numbers x between 3.97 and 4.03, including 3.97 and 4.03, we write it as \([3.97, 4.03]\). This gives us a clear and precise way to communicate the complete solution set in one go!

Solving Inequalities

Solving inequalities is similar to solving regular equations, but with some extra considerations. The inequality \(|x - 4| \leq 0.03\) requires us to find all values of \(x\) that make this statement true. Here's a step-by-step guide to solving such absolute value inequalities:

• First, understand the concept of absolute value: it represents the distance from zero. Here, \(|x - 4|\) represents how far \(x\) is from 4 on the number line.
• Translate the absolute value inequality into two separate inequalities. For \(|A| \leq B\), it becomes: \(A \leq B\) and \(A \geq -B\).
• Solve each inequality separately. In this problem, it becomes \(x - 4 \leq 0.03\) and \(x - 4 \geq -0.03\).
• Combine the solutions from each inequality to find the intersection. Both conditions must be satisfied at the same time.

By solving these inequalities, we found \(3.97 \leq x \leq 4.03\), which means any \(x\) within this range will satisfy the original inequality.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and subtraction). Understanding how to manipulate them is key to solving equations and inequalities. In the inequality \(|x - 4| \leq 0.03\), the algebraic expression \(x - 4\) is inside the absolute value bars. This represents a transformation of the variable \(x\) by subtracting 4. Dealing with expressions like these involves:

• Recognizing the operations involved: In this case, subtraction is the primary operation applied to \(x\).
• Reversing operations while solving: When you solve \(x - 4 \leq 0.03\), you add 4 to both sides to isolate \(x\), resulting in \(x \leq 4.03\).
• Working similarly with the opposite inequality \(x - 4 \geq -0.03\), results in \(x \geq 3.97\).

By practicing these operations, students can become adept at handling and simplifying algebraic expressions, which will greatly aid in solving both simple and complex mathematical problems.