Problem 107 - College Algebra Essentials

Question

When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.

Step-by-Step Solution

Answer

The set of all numbers that satisfy the given condition is \((- \infty, -1/3] \cup [3, \infty)\)

Step 1: Translate into algebraic expression

Let's denote the unknown number by \(x\). The statement '3 times a number is subtracted from 4' corresponds to the algebraic expression \(4 - 3x\). The statement 'the absolute value of the difference is at least 5' corresponds to the inequality \(|4 - 3x| \geq 5\). Putting these together, we get the algebraic inequality that we need to solve: \(|4 - 3x| \geq 5\).

Step 2: Solve the absolute value inequality

An absolute value inequality can be split into two separate inequalities. We know that either \(4 - 3x \geq 5\) or \(4 - 3x \leq -5\). Solving these separately gives \(x \leq -1/3\) from the first inequality and \(x \geq 3\) from the second inequality.

Step 3: Express the solution in interval notation

The solution \(x \leq -1/3\) corresponds to the interval \((- \infty, -1/3]\), and \(x \geq 3\) corresponds to the interval \([3, \infty)\). Since either one of these inequalities can be true for this problem, the union of these two intervals is the solution. This is written as \((- \infty, -1/3] \cup [3, \infty)\).

Key Concepts

Interval Notation

When it comes to understanding solutions to inequalities, one essential tool is interval notation. Interval notation is a shorthand used to describe sets of numbers that fall within certain bounds. This system is particularly handy when dealing with infinite sets or those that include an endpoint. For instance, the notation \( (-\infty, -1/3] \) represents all the numbers that are less than or equal to \( -1/3 \), and since there's no lower limit to how small these numbers can get, we symbolize this boundlessness with \( -\infty \).

Interval notation can also capture ranges without end, such as \( [3, \infty) \), where every number greater than or equal to \( 3 \) is included. Note that square brackets \( [ \) and \( ] \) indicate that the endpoint is included in the interval, while parentheses \( ( \) and \( ) \) suggest that the endpoint is not part of the set. In our example with absolute value inequalities, the union symbol 'cup' ( \( \cup \) ) joins two intervals, indicating that any number falling into either range satisfies the inequality.

Algebraic Expressions

Algebraic expressions form the basis of most algebra problems, and they are particularly crucial when representing complex mathematical relationships. A well-crafted algebraic expression, such as \( 4 - 3x \), can succinctly convey a wealth of information. In this instance, the expression represents the result of subtracting three times a number, \( x \), from \( 4 \).

Breaking Down Algebraic Expressions

Each part of an algebraic expression plays a role. The numeral \( 4 \) signifies a fixed value, while \( x \) stands in for an unknown quantity that we're solving for. The coefficient \( 3 \) preceding \( x \) amplifies the value of \( x \) by a factor of three. The subtraction between \( 4 \) and \( 3x \) indicates we're taking the latter away from the former. Understanding each component allows us to manipulate the expression confidently when solving complex equations or inequalities.

Inequality Solutions

Finding solutions to inequalities, particularly absolute value inequalities, requires a systematic approach. The inequality \( |4 - 3x| \geq 5 \) poses a unique challenge because the absolute value function dictates that the expression inside the bars must be either greater than or equal to \( 5 \) or less than or equal to \( -5 \) for the inequality to hold.

Essentially, you're dealing with two separate scenarios. By solving both \( 4 - 3x \geq 5 \) and \( 4 - 3x \leq -5 \) independently, you find two sets of solutions which, when combined, give the complete solution to the original inequality. Inequalities often have multiple valid answers, and it's important to consider all possibilities to find a comprehensive solution set. This approach leads to a clear understanding of the range of values that satisfy the initial conditions.