Problem 63 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ |2 x-6|<8 $$

Step-by-Step Solution

Answer

The solution to the absolute value inequality \(|2x-6|<8\) is \( -1 < x < 7\).

Step 1: Rewrite the inequality

Rewrite the given inequality as \( -8 < 2x - 6 < 8\). This means that the value of \(2x - 6\) lies between -8 and 8.

Step 2: Solve for x each part of the inequality

Solve each part of the inequality to isolate \(x\). The part \( -8 < 2x - 6\) becomes \( -8 +6 < 2x \), which simplifies to \( -2 < 2x \) and finally to \( -1 < x \). The part \(2x - 6 < 8\) becomes \( 2x < 8 + 6\), which simplifies to \( 2x < 14 \) and finally to \( x < 7 \). Combine these two to get the final solution.

Step 3: Write the final solution

The final solution of the inequality is \( -1 < x < 7\) which means \(x\) is greater than -1 and less than 7.

Key Concepts

Inequality Solving

When solving inequalities, the aim is to find the set of values that satisfy the inequality. The approach is similar to solving equations, but with some key differences. Consider the inequality \( |2x - 6| < 8 \). Absolute value inequalities like this one require us to consider both the positive and negative scenarios. This is because the absolute value function measures the distance from zero, regardless of direction.

To solve \( |2x - 6| < 8 \), we rewrite it as \(-8 < 2x - 6 < 8\). This creates a compound inequality, which represents that the expression \(2x - 6\) is confined between -8 and 8. Solving this compound inequality involves handling both sides, which allows us to isolate \(x\) and find the range of values that \(x\) can take.

Always remember that while performing operations, such as adding or subtracting across inequalities, the direction of the inequality symbol remains the same. However, flipping the direction is required when multiplying or dividing by a negative number.

Algebra

Algebra involves finding unknown values through manipulation of algebraic expressions and equations. In solving \( |2x - 6| < 8 \), algebra helps us break down and solve the individual parts of the compound inequality step by step. Let's break down the steps involved:

• Start with solving the left part: \(-8 < 2x - 6\).
• To isolate \(2x\), first add 6 to both sides, resulting in \(-2 < 2x\).
• Divide by 2 to solve for \(x\), yielding \(-1 < x\).

• Next, tackle the right part: \(2x - 6 < 8\).
• Similarly, add 6 to both sides to get \(2x < 14\).
• Divide by 2, giving us \(x < 7\).

These calculations show how algebra is used efficiently to solve the inequality by isolating the variable \(x\). The solution \(-1 < x < 7\) tells us where \(x\) resides on the number line, thereby solving the inequality.

Absolute Value Properties

Absolute value measures the distance of a number from zero. Its properties are fundamental when dealing with equations or inequalities that involve the absolute value, such as \( |2x - 6| < 8 \). Understanding these properties helps break down complex expressions into manageable pieces.

The absolute value can introduce situations with dual possibilities, hence rewriting the inequality \(|2x - 6| < 8\) as \(-8 < 2x - 6 < 8\). This is possible because the absolute value function \(|A| < B\) implies \(-B < A < B\).

Here are a few important points about absolute value properties:

• If \(|A| < B\), then \(-B < A < B\).
• If \(|A| > B\), then \(A < -B\) or \(A > B\).

These properties assist in converting absolute value inequalities into compound inequalities, simplifying the process of finding solutions. By understanding these concepts, solving such inequalities becomes a straightforward process that can be systematically approached.