Problem 29 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ |2 x-3| \leq 0.4 $$

Step-by-Step Solution

Answer

\([1.3, 1.7]\)

Step 1: Understand the Absolute Value Inequality

The inequality \(|2x - 3| \leq 0.4\) means that the expression inside the absolute value, \(2x - 3\), lies between \(-0.4\) and \(0.4\). Mathematically, this is written as \[-0.4 \leq 2x - 3 \leq 0.4\] This represents a compound inequality that we need to solve.

Step 2: Solve the Left Part of the Inequality

Focus on the first part of the compound inequality: \[-0.4 \leq 2x - 3\]To isolate \(x\), first add 3 to both sides:\[2.6 \leq 2x\]Next, divide each side by 2:\[1.3 \leq x\]

Step 3: Solve the Right Part of the Inequality

Now, solve the right part of the compound inequality: \[2x - 3 \leq 0.4\]Add 3 to both sides:\[2x \leq 3.4\]Then, divide each side by 2:\[x \leq 1.7\]

Step 4: Combine the Results

Combine the solutions of both parts, from Step 2 and Step 3: \[1.3 \leq x \leq 1.7\]This combined inequality shows the range of values that \(x\) can take.

Step 5: Express the Solution in Interval Notation

The solution \(1.3 \leq x \leq 1.7\) in interval notation is written as:\[[1.3, 1.7]\]The brackets indicate that the endpoints \(1.3\) and \(1.7\) are included in the solution.

Key Concepts

Compound Inequality

The concept of a compound inequality involves combining two inequalities into one statement. When you encounter a compound inequality like \(-0.4 \leq 2x - 3 \leq 0.4\), it signifies that the expression \(2x - 3\) must satisfy both conditions simultaneously.

This means that the expression lies between two given numbers—in this case, between \(-0.4\) and \0.4\.

• Compound inequalities are often connected by the word "and," as both parts of the inequality must be true at the same time.
• You solve them by addressing each part separately and then combining the results.

Addressing both parts properly ensures you capture the entire solution space for the variable \(x\).

Interval Notation

Interval notation is a way to succinctly express a range of numbers, particularly useful in solutions to inequalities. In the solution \[1.3 \leq x \leq 1.7\], interval notation provides a compact representation: \[1.3, 1.7\].

This means that the variable \(x\) takes on all values from \(1.3\) to \(1.7\), inclusive of the endpoints.

• The square brackets \[ \, , \, \] indicate inclusion, meaning \(x\) can be equal to the endpoints: \(1.3\) and \(1.7\) in this context.
• If the solution excluded endpoints, parentheses \( \, , \, \) would be used. However, since our original inequality includes both endpoints, square brackets are appropriate.

Interval notation provides clarity and simplicity in communicating solution sets, especially when dealing with continuous data on the number line.

Solving Inequalities

Solving inequalities involves finding all values of the variable that satisfy the inequality statement. It's similar to solving equations, but with some key differences. Here’s a quick overview:

1. **Reverse Operations**: You perform operations similar to solving equations—adding, subtracting, multiplying, or dividing both sides—keeping track of operations that can change the inequality sign (particularly multiplication or division by a negative number).

• In our exercise, we solved \(-0.4 \leq 2x - 3 \) by first adding 3 to both sides, then dividing by 2.
• Similarly, for \[2x - 3 \leq 0.4\], we used addition and division to isolate \(x\).

2. **Maintain Inequality Direction**: Ensure the direction of the inequality remains correct throughout the operations, unless modifying by a negative factor.3. **Combine Results**: If dealing with a compound inequality, combine the results to identify the full range of solutions.

• This is done by finding overlap (or intersection) of solutions from individual inequalities, as we combined \1.3 \leq x\ and \x \leq 1.7\.

The process culminates in expressing solutions, often in interval notation, to demonstrate all permissible values of the variable. Remember to be attentive to keep the solution valid throughout!