Problem 55 - A Graphical Approach to College Algebra

Question

Solve each equation or inequality. $$|15-x|<7$$

Step-by-Step Solution

Answer

\(8 < x < 22\)

Step 1: Understand Absolute Value Inequality

The inequality \(|15-x| < 7\) involves an absolute value, which means we are looking for the set of all \(x\) values that keep the expression inside the absolute value in a range less than 7 units from zero. This gives us two scenarios to consider.

Step 2: Set Up Two Inequalities

For an absolute value inequality \(|A| < B\), we can break it down into two parts: \(A < B\) and \(A > -B\). In this case, \(A\) is \(15-x\), so we set up the equations:1. \(15-x < 7\) 2. \(15-x > -7\)

Step 3: Solve First Inequality

Solve the first inequality \(15-x < 7\):1. Subtract 15 from both sides: \(-x < 7 - 15\)2. Simplify: \(-x < -8\)3. Multiply both sides by -1 and reverse the inequality: \(x > 8\).

Step 4: Solve Second Inequality

Solve the second inequality \(15-x > -7\):1. Subtract 15 from both sides: \(-x > -7 - 15\)2. Simplify: \(-x > -22\)3. Multiply both sides by -1 and reverse the inequality: \(x < 22\).

Step 5: Combine Solutions

Combine the results of the two inequalities. The solution is where the inequalities overlap: \(x > 8\) and \(x < 22\). This can be written as: \\(8 < x < 22\).

Step 6: Verify the Solution

We can check the boundary values, \(x = 8\) and \(x = 22\), to ensure they do not satisfy the original inequality. Evaluate at \(x = 10\), which is within the solution set: \(|15-10| = 5\), which is less than 7. This confirms that our solution \(8 < x < 22\) is correct.

Key Concepts

Inequality Solutions

When tackling inequality problems like \(|15-x| < 7\), it’s important to understand they involve finding a range of values rather than a single solution. Inequalities tell us where expressions are larger or smaller than certain values. For absolute value inequalities, we need to address the particular nature of absolute values needing special attention.

• Absolute inequalities can typically be handled by turning them into two separate linear inequalities.
• Once split, solve each inequality independently to identify potential solution ranges for the variable in question.

It's essential to look at the overlap of these solution sets. Only the overlapping range will work in satisfying the conditions of the given inequality. In our example, we solved for \( x > 8 \) and \( x < 22 \) to conclude that the solution set is \( 8 < x < 22 \). Breaking it into smaller steps makes the solution approachable.

Absolute Value Properties

Absolute value expressions reflect the magnitude of a number, irrespective of its sign. In simple terms, the absolute value of a number is always positive. This characteristic plays a crucial role in understanding how to handle absolute value inequalities.

• The expression \(|A| < B\) means the value inside the absolute brackets, \(A\), is less than \(B\) units away from zero in both directions on the number line.
• Consequently, it always results in two inequalities: one dealing with the positive side and another with the negative side of the spectrum.

This approach helps in bounding the expressions within a range, resulting in potential solution intervals where both conditions are satisfied. This means that any value of \(x\) we find within this interval will keep the expression \(15-x\) less than 7 units away from zero.

Algebraic Inequalities

Algebraic inequalities require careful manipulation to find solutions that respect the inequality signs. Here are some steps to help solve them efficiently:

• Recognize and set up the inequalities: Convert inequalities involving absolute values into two separate linear inequalities as shown with \(15-x < 7\) and \(15-x > -7\).
• Solve each inequality: Use basic algebraic techniques like adding, subtracting, multiplying, or dividing to isolate the variable on one side of the inequality.
• Remember to reverse the inequality sign when multiplying or dividing by a negative number, as seen in both inequalities while solving for \(-x\).

After solving, always combine the results critically, looking for intersections that fit within both conditions. This results in a solution set representing the acceptable values for \(x\). Understanding these basic principles of handling algebraic inequalities ensures accuracy and efficiency in solving similar problems.