Problem 65 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Write the solution in interval notation. $$|0.25 x-1|>3$$

Step-by-Step Solution

Answer

(-∞, -8) ∪ (16, ∞)

Step 1: Understand the Definition of Absolute Value Inequality

The inequality \(|0.25x - 1| > 3\) means that the quantity \(0.25x - 1\) is either greater than 3 or less than -3.

Step 2: Break The Absolute Value Inequality into Two Inequalities

Based on our understanding of absolute value, we break it down into two separate inequalities: \(0.25x - 1 > 3\) and \(0.25x - 1 < -3\). These represent the two scenarios where the expression inside the absolute value is either greater than 3 or less than -3.

Step 3: Solve the First Inequality

Solve \(0.25x - 1 > 3\):1. Add 1 to both sides to isolate the term with \(x\): \[0.25x > 4\]2. Divide both sides by 0.25 to solve for \(x\): \[x > 16\]

Step 4: Solve the Second Inequality

Solve \(0.25x - 1 < -3\):1. Add 1 to both sides: \[0.25x < -2\]2. Divide both sides by 0.25: \[x < -8\]

Step 5: Write the Solution in Interval Notation

The solutions to the inequalities \(x > 16\) and \(x < -8\) are disjoint intervals. In interval notation, this is written as the union of two intervals: \((-fty, -8) \cup (16, fty)\).

Key Concepts

Understanding Absolute Value

The concept of absolute value is crucial when dealing with inequalities like \(|0.25x - 1| > 3\). Absolute value can be thought of as the distance of a number from zero on a number line.
It's always non-negative, meaning it doesn’t matter if the number inside the absolute value is positive or negative; the result will always be a positive or zero.

• An expression inside absolute value can be zero, positive, or negative.
• The equation breaks into two different inequalities without the absolute value.

In the given inequality, \(|0.25x - 1| > 3\), we're essentially trying to find all values of \(x\) such that \(0.25x - 1\) is either greater than 3 or less than -3.
Understanding this allows us to split the original inequality into two simpler inequalities: \(0.25x - 1 > 3\) and \(0.25x - 1 < -3\). Solving these will provide the intervals where the original inequality holds true.

Interval Notation

Interval notation is a handy way of representing sets of numbers, particularly solutions to inequalities. It uses brackets and parentheses to denote intervals on the number line.
For our specific inequality, once we solve it, the answers are expressed using interval notation.

• Parentheses \((\) or \(()\) are used to show that a number is not included in the interval, like an open end.
• Brackets \([\) or \([]\) indicate that the number is included, like a closed end.

For example, the solution \(x > 16\) is written in interval notation as \((16, \infty)\). Similarly, \(x < -8\) is expressed as \((-\infty, -8)\).
The final solution to our inequality is the union of these two intervals: \((-\infty, -8) \cup (16, \infty)\). This signifies that \(x\) can be any number less than -8 or greater than 16.

Solving Inequalities

Solving inequalities involving absolute value can initially seem complex, but they are straightforward once broken into their components.
Let's consider our inequality, \(|0.25x - 1| > 3\). Here are the steps to solve absolute value inequalities effectively:

• Break the inequality into two separate inequalities. This accounts for both primary conditions of absolute values.
• Solve each inequality individually. Treat them just like regular equations.

For instance, to solve \(0.25x - 1 > 3\), add 1 to both sides to isolate the term \(0.25x\), resulting in \(0.25x > 4\).
Then divide by 0.25 to solve for \(x\), giving us \(x > 16\).
Similarly, for \(0.25x - 1 < -3\), add 1 to both sides resulting in \(0.25x < -2\), and then divide by 0.25 to find \(x < -8\).
By approaching inequalities step-by-step and using these techniques, you can confidently solve absolute value inequalities and express your answers in interval notation.