Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the indicated implication is true. $$ |x-3|<0.5 \Rightarrow|5 x-15|<2.5 $$

Step-by-Step Solution

Answer

The implication is true; both conditions hold for \(2.5 < x < 3.5\).

Step 1: Understand the Implication

First, we need to understand what is being asked. The implication \(|x-3|<0.5 \Rightarrow|5x-15|<2.5\) means that whenever \(|x-3|<0.5\) is true, it should logically lead to \(|5x-15|<2.5\) being true.

Step 2: Express Absolute Value as Inequalities

The inequality \(|x-3|<0.5\) can be expanded to the compound inequality \(-0.5 < x - 3 < 0.5\). This can be solved to find the range of \(x\) values that satisfy this inequality.

Step 3: Solve for x

Solving the inequality \(-0.5 < x - 3 < 0.5\), add 3 to each part:\[-0.5 + 3 < x < 0.5 + 3\]\[2.5 < x < 3.5\]Therefore, the values of \(x\) that satisfy \(|x-3|<0.5\) are all those in the interval \(2.5 < x < 3.5\).

Step 4: Analyze the Consequence of x Values

We now need to check if these same \(x\) values will satisfy \(|5x-15|<2.5\). We express it as a compound inequality: \(-2.5 < 5x - 15 < 2.5\).

Step 5: Solve the Consequence Inequality

Solving \(-2.5 < 5x - 15 < 2.5\), add 15 to each part:\[-2.5 + 15 < 5x < 2.5 + 15\]\[12.5 < 5x < 17.5\]Now, divide the entire inequality by 5:\[2.5 < x < 3.5\]This interval is exactly the same as the result from Step 3.

Step 6: Conclusion

Since the solutions of \(|x-3|<0.5\) exactly match those of \(|5x-15|<2.5\), this proves that the implication is true. Whenever \(|x-3|<0.5\), \(|5x-15|<2.5\) will always be satisfied.

Key Concepts

Inequalities

Inequalities are mathematical expressions that indicate the relative size or order of two values. Instead of stating that two expressions are equal, they show whether one is greater than, less than, or equal to another value. Consider the inequality \(-0.5 < x - 3 < 0.5\). This means that the value inside the absolute value is between -0.5 and 0.5.

• This type of inequality is known as a compound inequality because it combines two inequalities into a single statement.
• The goal with inequalities is often to isolate the variable to find its range of possible values.

When solving inequalities, the direction of the inequality sign remains unchanged unless you multiply or divide both sides of the inequality by a negative number. In this exercise, solving the inequality helped us find that \(x\) must lie between 2.5 and 3.5 in order to satisfy \(|x-3|<0.5\). This idea can be extended to understand how others, like the implications shown here, might behave.

Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. Thus, the absolute value is always non-negative. The expression \(|x-3|<0.5\) assesses how close the variable \(x\) is to the number 3.

• The purpose of using absolute values in inequalities is to ensure that we consider both positive and negative deviations from a central point.
• In the context of this exercise, it helped define a tolerance zone around 3, within which \(x\) can fluctuate.

To transform an absolute value inequality into a form that is easier to manage, like the compound inequality \(-0.5 < x - 3 < 0.5\), enables solving for \(x\) directly. Understanding absolute value is crucial for managing bounds and finding solutions to problems involving distances.

Implication Proof

An implication proof demonstrates that if the initial condition is true, then the outcome also must be true. In logical terms, it is denoted by a statement such as \(A \Rightarrow B\). For this exercise, \(|x-3|<0.5 \Rightarrow |5x-15|<2.5\) means if the first inequality is satisfied, then the second one must also hold true.

• This requires proving that whenever any number within the bounds of the first inequality is selected, it will also satisfy the second inequality.
• We achieved this by solving each inequality separately and demonstrating that they lead to the same range of \(x\).

Such proof promotes rigorous logical reasoning, confirming the relationships between inequalities of absolute values.

Mathematical Logic

Mathematical logic is a subfield of mathematics exploring formal systems and their applications. It forms the backbone of proving implications and other mathematical relationships. In this problem, the logical principle of implications has been applied to prove that one statement leads to another.

• A sound approach to logic is necessary here since exercises like this depict conditional scenarios—one condition triggers another.
• Logical connectors, like "implies" \((\Rightarrow)\), are crucial in forming such relationships.

Understanding and practicing mathematical logic allows students to follow these arguments and develop powerful reasoning skills. This is critical for complex problem-solving not only in mathematics but across various disciplines.