Problem 85 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ 3 \leq|2 x-1| $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 2\) or \(x \leq -1\).

Step 1: Express the Inequality with Positive Expression

The absolute value can be expressed positively as: \(3 \leq 2x - 1\).

Step 2: Solve the Positive Inequality

Isolate the variable on one side. Add 1 to both sides to get: \(2x \geq 4\), and then divide by 2 to get: \(x \geq 2\).

Step 3: Express the Inequality with Negative Expression

The absolute value can also be expressed negatively, which leads to a different inequality: \(3 \leq -(2x - 1)\).

Step 4: Solve the Negative Inequality

This simplifies further to \(3 \leq -2x + 1\). Subtract 1 from both sides to get \(2 \leq -2x\), and then divide by -2 (which will reverse the inequality), to get \(x \leq -1\).

Key Concepts

Solving Absolute Value Inequalities

When you're tackling absolute value inequalities, the key is to consider the definition of absolute value: it's the distance from zero on the number line. The absolute value of any number or expression is always non-negative. Hence, solving these inequalities requires you to break down the expression into two separate inequalities based on positive and negative scenarios.Let's explore this idea with the given exercise. The inequality \(3 \leq|2x-1|\) can be interpreted in two different ways, due to the dual nature of absolute values. This means:

• \(2x-1\) might be greater than or equal to 3: \(2x - 1 \geq 3\).
• Or, \(2x-1\) might be less than or equal to -3: \(2x - 1 \leq -3\).

These two possibilities form a compound inequality, which you'll need to solve separately to determine the full set of solutions.

Inequality Solutions Explained

After breaking down the absolute value inequality into two simple inequalities, you can solve each one independently. This provides a clearer path to the solution.For the first part \(2x - 1 \geq 3\), perform the following:

• Add 1 to both sides: \(2x \geq 4\).
• Divide by 2 to isolate \(x\): \(x \geq 2\).

For the second part \(2x - 1 \leq -3\), it becomes:

• Add 1 to both sides: \(2x \leq -2\).
• Divide by 2 to isolate \(x\): \(x \leq -1\).

Combining these solutions, we find that our variable \(x\) can be any value less than or equal to -1 or greater than or equal to 2. These are the intervals where the original inequality holds true. This methodical approach ensures you're covering all scenarios by working through each simpler inequality individually.

Understanding Absolute Value Expressions

An absolute value expression involves bars around a number or variable, like \(|2x-1|\). This specific notation dictates that we consider both the positive and negative potential of the expression inside the bars.For example, the inequality \(3 \leq |2x-1|\) suggests looking at two possible cases:

• \(2x-1\) could have a positive solution, leading to \(2x-1 \geq 3\).
• \(2x-1\) could have a negative solution, expressed as \(2x-1 \leq -3\) when you solve its opposite form.

This duality is a hallmark of absolute value situations and is essential in broadening our understanding of how distance works in a mathematical context, always focusing on exploring both 'directions' or scenarios.

The Role of Mathematical Reasoning

Mathematical reasoning is about forming logical deductions from the given expressions. When dealing with absolute value inequalities, reasoning is used to understand the nature of the solution set.First, you need to recognize that absolute value always results in a non-negative number. This informs your approach since you're required to consider 'both sides' of the expression.
This involves more than just solving equations; it requires:

• Interpreting the meaning of the inequality in the context of absolute value.
• Judging which values of \(x\) satisfy the inequality.
• Checking if the inequalities flip when multiplied or divided by a negative number.

Adopting this reasoning ensures that you not only perform the correct operations but understand the implications of each step. It is this critical thinking skill that ultimately enables you to solve such inequalities with confidence and precision.