Problem 56 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Write the solution in interval notation. $$|15-x|<7$$

Step-by-Step Solution

Answer

The solution is \((8, 22)\).

Step 1: Set Up the Inequality

To solve the inequality \(|15-x|<7\), we understand that it represents the range of values of \(x\) such that the distance from 15 is less than 7. We can split it into two separate inequalities: \(15-x < 7\) and \(15-x > -7\).

Step 2: Solve the First Inequality

Solve the inequality \(15-x < 7\). Subtract 15 from both sides to get:\[-x < 7 - 15\]\[-x < -8\]Multiply by -1 and reverse the inequality sign:\[x > 8\]

Step 3: Solve the Second Inequality

Now solve \(15-x > -7\). Subtract 15 from both sides:\[-x > -7 - 15\]\[-x > -22\]Multiply by -1 and reverse the inequality sign:\[x < 22\]

Step 4: Combine Solutions

Combine the solutions from Step 2 and Step 3 to find the values of \(x\) that satisfy both. We have \(x > 8\) and \(x < 22\).

Step 5: Express in Interval Notation

Write the solution in interval notation as the values between 8 and 22, not including the endpoints:\((8, 22)\).

Key Concepts

Understanding Absolute Value

The concept of absolute value is a fundamental aspect of algebra. It refers to the distance of a number from zero on the number line. The absolute value of a number is always non-negative. It is symbolized by vertical bars, like \( |x|\). For instance, \( |3| = 3\), and \( |-3| = 3\) because both numbers are three units away from zero.

When dealing with absolute value inequalities, such as \( |15-x| < 7\), it's important to interpret this correctly. It implies we are looking for all the possible values of \( x\) such that when 15 is subtracted, the result is within 7 units of zero. This leads to two conditions. One where the inequality is written as \( 15-x < 7\) and another as \( 15-x > -7\).

This splitting allows us to handle the positive and negative scenarios of the absolute value, ensuring we consider all valid values of \( x\).

Interval Notation

Interval notation is a concise way of expressing a range of numbers, often used to represent the solution of inequalities. In this system, brackets and parentheses denote whether endpoints are included or excluded.

• Round brackets, \( ()\), mean the number is not included (open interval).
• Square brackets, \[ []\], mean the number is included (closed interval).

For example, if we have a solution \( (8, 22)\), it means all numbers greater than 8 and less than 22 satisfy our inequality, but 8 and 22 themselves do not. This "open interval" notation is ideal for representing ranges that do not include their boundaries.

By practicing interval notation, students gain proficiency in expressing their answers in a format that's both universal and respected in mathematical contexts.

Key Algebra Concepts

Several core algebra concepts play a pivotal role in solving inequalities, specifically when absolute values and interval notations are involved:

• Solving Linear Inequalities: This involves manipulating an equation to isolate the variable on one side. As shown in our example problem, operations like adding, subtracting, or multiplying affect the direction of inequality signs.
• Reversing Inequality Signs: Any time an inequality is multiplied or divided by a negative number, the inequality sign flips direction. This rule is crucial, such as when resolving \( -x < -8\) to \( x > 8\).

Let’s not forget that understanding these steps allows us to integrate algebraic manipulations with logical reasoning to resolve complex mathematical questions. Building these skills prepares students for the logical structures they'll encounter in higher-level mathematics.