Problem 108 - Algebra and Trigonometry

Question

When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. 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 \([2, +\infty)\).

Step 1: Express the given condition as an inequality

Let x be the number in question. According to the task, we have: \(|5 - 4x| \leq 13\). This is the inequality that needs to be solved.

Step 2: Break the absolute value inequality into two separate inequalities

An absolute value inequality \(|a| \leq b\) can be broken down into two separate inequalities: \(-b \leq a \leq b\). Hence, the inequality \(|5 - 4x| \leq 13\) becomes \(-13 \leq 5 - 4x \leq 13\).

Step 3: Solve the two inequalities

Now, we need to solve the inequalities \(-13 \leq 5 - 4x\) and \(5 - 4x \leq 13\). Solving the first inequality gives: \(-13 + 5 \leq -4x \rightarrow -8 \leq -4x \rightarrow x \geq 2\). Similarly, the second inequality gives: \(5 - 13 \leq -4x \rightarrow -8 \leq -4x \rightarrow x \geq 2\).

Step 4: Present the solution as an interval

The solution to the inequality is \(x \geq 2\). In interval notation, this is represented as \([2, +\infty)\). This represents all values greater than or equal to 2.

Key Concepts

Interval Notation

Interval notation is a way of writing the set of solutions for an inequality. It uses brackets to clearly show where a solution set begins and ends. For example, when a solution includes a particular number, square brackets \( [ ] \) are used, meaning the endpoint is included in the set. If the endpoint is not included, parentheses \( ( ) \) are used instead.

In the context of our problem, once we solve the inequality \( 4x - 5 \leq 13 \), we find that \( x \geq 2 \). This means x can be 2 or any number greater than 2.

• The expression \( [2, +\infty) \) means all numbers from 2 up to infinity, including 2.
• \( +\infty \) always uses parentheses because infinity is not a number we can reach or include.

Using interval notation helps simplify the representation of the solution and makes it easier for others to understand.

Solving Inequalities

Solving inequalities involves finding all values of a variable that make an inequality true. When dealing with absolute value inequalities, the process involves a few extra steps compared to regular inequalities.

In the original exercise, we dealt with the inequality \( |5 - 4x| \leq 13 \). The absolute value symbol affects how the inequality is handled. Here are the steps to solve such inequalities:

• Write the inequality without the absolute value: \( -13 \leq 5 - 4x \leq 13 \).
• This breaks the original inequality into two simpler inequalities: \( -13 \leq 5 - 4x \) and \( 5 - 4x \leq 13 \).
• Solve each of these simpler inequalities to find the range of possible values for x.

A key point to remember is that flipping the inequality sign happens when multiplying or dividing by a negative number. Simplifying these inequalities gives us the range for x, which can then be expressed using interval notation.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (such as addition, subtraction, multiplication, and division). Understanding these expressions is crucial when dealing with equations and inequalities.

In our problem, the expression \( |5 - 4x| \leq 13 \) combines numbers, a variable (x), and operations. Here's how to interpret it:

• The \( 5 - 4x \) part represents a linear equation where x is the variable.
• The absolute value symbol \( | | \) indicates the distance from zero, eliminating any negative signs.
• The inequality sign \( \leq \) specifies that the expression on the left must be less than or equal to 13.

Manipulating algebraic expressions by performing the same operation on both sides is essential in solving inequalities. This approach ensures we maintain balance and accurately isolate the variable to find its possible values.