Problem 106 - A Graphical Approach to Precalculus with Limits

Question

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=7-x\) must be less than 0.002 unit from 1.2.

Step-by-Step Solution

Answer

5.798 < x < 5.802.

Step 1: Identify the Condition

The condition states that the value of \( f(x) = 7 - x \) must be less than 0.002 units from 1.2. This can be mathematically expressed in terms of the absolute value as: \( |f(x) - 1.2| < 0.002 \).

Step 2: Substitute f(x) = 7 - x

Substitute the function \( f(x) = 7-x \) into the inequality: \( |(7-x) - 1.2| < 0.002 \). This simplifies to \( |5.8 - x| < 0.002 \).

Step 3: Remove the Absolute Value

The inequality \( |5.8 - x| < 0.002 \) can be expressed without absolute value by creating a compound inequality: \( -0.002 < 5.8 - x < 0.002 \).

Step 4: Solve the Inequality for x

We solve the compound inequality from the previous step:1. For \( -0.002 < 5.8 - x \): - Add \( x \) to both sides: \( -0.002 + x < 5.8 \) - Then, add 0.002 to both sides: \( x < 5.802 \).2. For \( 5.8 - x < 0.002 \): - Subtract 5.8 from both sides: \( -x < -5.798 \) - Multiply by -1 (and remember to flip the inequality): \( x > 5.798 \).Thus, the solution is \( 5.798 < x < 5.802 \).

Key Concepts

Compound Inequality

A compound inequality is a combination of two or more inequalities that are joined together by the words "and" or "or." In the context of this exercise, the compound inequality is formed when we remove the absolute value from the inequality \(|5.8 - x| < 0.002\). This results in two separate inequalities:

• First inequality: \(-0.002 < 5.8 - x\)
• Second inequality: \(5.8 - x < 0.002\)

The use of "and" implies that both conditions must be met simultaneously. To find the possible values for \(x\), we need to solve both inequalities and identify the overlapping values. This combination ensures \(x\) is just right—neither too low nor too high—which fits within a tiny range defined by the original condition.

Inequality Solving

Solving inequalities is a lot like solving equations, but with extra attention to detail. In this particular problem, we start solving the compound inequality \(-0.002 < 5.8 - x < 0.002\). Let's see how it's done step by step:
For \(-0.002 < 5.8 - x\):

• First, add \(x\) to both sides to obtain: \(-0.002 + x < 5.8\).
• Next, add 0.002 to both sides to finalize: \(x < 5.802\).

For \(5.8 - x < 0.002\):

• Subtract 5.8 from both sides: \(-x < -5.798\).
• Multiply both sides by -1 (and remember to reverse the inequality sign): \(x > 5.798\).

Once both are solved, the solutions get combined to give the range of \(x\): \(5.798 < x < 5.802\). This process highlights careful manipulation of the inequality symbols, especially when multiplying or dividing by negative numbers, to find the true solution set.

Mathematical Expressions

Mathematical expressions are like phrases in a sentence—they convey precise instructions and information. In this exercise, the expression \(f(x) = 7 - x\) is used to form an absolute value inequality that ensures \(f(x)\) stays within a small distance from a target value. The mathematical expression \(|f(x) - 1.2| < 0.002\) demands that when you plug in any acceptable value of \(x\), the output of \(f(x)\) doesn't stray more than 0.002 units away from 1.2. This precision is expressed initially through the absolute value, which measures how far a number is from zero irrespective of direction, and is then translated into a range with the aid of inequalities, giving us a powerful tool to define specific constraints in mathematical terms.