Problem 107 - Algebra and Trigonometry

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 numbers that satisfy this condition are in the intervals \((- \infty, -1/3] \cup [3, \infty)\).

Step 1: Write the problem as an absolute value inequality

First, the problem is written as an absolute value inequality. The phrase '3 times a number is subtracted from 4' can be written as \(4 - 3x\). This is at least 5 units from zero, so the problem can be written as: \(|4 - 3x| \geq 5\).

Step 2: Solve the inequality

Absolute value inequalities are actually two different inequalities. We solve for both the positive and negative values. This means the equation is split into two: \(4 - 3x \geq 5\) and \(4 - 3x \leq -5\). Then after rearranging these become \(3x \leq -1\) and \(3x \geq 9\), and then dividing by 3 for both give us \(x \leq -1/3\) and \(x \geq 3\).

Step 3: Write the solution in interval notation

Interval notation is a way to represent a set of real numbers. The solutions \(x \leq -1/3\) and \(x \geq 3\) become \((- \infty, -1/3]\) and \([3, \infty)\), respectively. The union symbol is used to connect these two intervals, showing that the solution can be any number from either interval.

Key Concepts

Interval Notation

Interval notation is a concise way to express a range of numbers that satisfy certain conditions, such as solutions to inequalities. It uses brackets and parentheses to describe subsets of the real number line.

Here's a quick guide on how to read and write interval notation:

• Brackets, like \([\ ]\), are used when a number is included in the interval, known as a "closed interval." For example, \([3, \infty)\) means 3 is included in the range, but infinity is not.


• Parentheses, like \((\ )\), denote a "open interval," where the number is not included. For instance, \((-\infty, -1/3]\) includes all numbers less than or equal to -1/3.


• The union symbol \((\cup)\) joins different intervals, indicating that numbers in either interval are part of the solution.

The expression \((-\infty, -1/3] \cup [3, \infty)\) implies that the solution includes numbers less than or equal to -1/3 and numbers greater than or equal to 3.

Real Numbers

Real numbers encompass all the numbers you typically encounter in everyday math and beyond. They include all the rational numbers such as integers and fractions, as well as all irrational numbers, those that cannot be expressed as simple fractions like \(\pi\) or the square root of 2.

In this context, we are interested in solutions that are part of the real number set. Real numbers are important because they help us solve practical problems, like the absolute value inequality in this exercise. When we use interval notation, we're describing portions of the real number line where the solutions lie.

Because real numbers cover both rational and irrational numbers in an unbroken sequence without gaps, the solutions for exercises like these often encompass broad and continuous sets, meant to include virtually any conceivable number without exception.

Algebraic Solutions

Solving absolute value inequalities with algebra involves a few key steps, mostly centered on breaking down the inequality into simpler parts. Here's how you can manage it:

First, recognize that absolute value inequalities, like \(|4 - 3x| \geq 5\), always translate into two linear inequalities:

• Positive inequality: \(4 - 3x \geq 5\)


• Negative inequality: \(4 - 3x \leq -5\)

Begin by solving each of these separately:
- Rearrange each inequality to isolate \(x\): - From \(4 - 3x \geq 5\), we subtract 4 from both sides and then divide by -3, flipping the inequality sign to get \(x \leq -1/3\).
- From \(4 - 3x \leq -5\), follow a similar process to find \(x \geq 3\).

Finally, these separate solutions are typically pieced together via union of the intervals, which showcases all the possible values that \(x\) might take, forming the complete algebraic solution in the context of real numbers.