Problem 85 - College Algebra

Question

Solve absolute value inequality. \(3 \leq|2 x-1|\)

Step-by-Step Solution

Answer

The solution for the absolute value inequality \(3 \leq |2x - 1|\) is all values of \(x\) which satisfy \(x \leq -1\) or \(x \geq 2\).

Step 1: Divide the inequality into two separate inequalities

An absolute value inequality like \(3 \leq |2x - 1|\) can be divided into two separate cases: Case 1: When \(2x - 1\) is positive, then \(2x - 1 \geq 3\).Case 2: When \(2x - 1\) is negative, then \(-(2x - 1) \geq 3\).

Step 2: Solve each inequality

We solve both inequalities separately.Case 1: \(2x - 1 \geq 3\) gives \(2x \geq 4\) and therefore \(x \geq 2\).Case 2: \(-(2x - 1) \geq 3\) gives \(-2x + 1 \geq 3\), therefore \(-2x \geq 2\) and hence \(x \leq -1\).

Step 3: Union of solutions

The solutions for the original absolute value inequality are the numbers that satisfy either of the inequalities we solved for in Step 2. That means our solution is the union of \(x \geq 2\) and \(x \leq -1\). Therefore, the solution of the given absolute value inequality \(3 \leq |2x - 1|\) is all \(x\) such that \(x \leq -1\) or \(x \geq 2\).

Key Concepts

Understanding Inequalities

Inequalities are an essential concept in algebra. They tell us how one value is related to another, using symbols to express this relationship. There are several inequality symbols:

• "<" means "less than".
• ">" means "greater than".
• "≤" means "less than or equal to".
• "≥" means "greater than or equal to".

In algebra, inequalities show a range of possible solutions instead of one specific answer. When solving inequalities, you follow similar steps to solving equations. However, there's a crucial rule: if you multiply or divide an inequality by a negative number, you must flip the inequality sign. This occurs because multiplying or dividing by a negative changes the direction of the inequality.
While equations have unique solutions, inequalities often have multiple solutions that form an interval on the number line. In our specific example, we are looking for all the values of \(x\) that satisfy the inequality in the context of an absolute value.

The Concept of Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It’s always a non-negative value. The symbol \(|a|\) denotes the absolute value of \(a\).

• If \(a \geq 0\), then \(|a| = a\).
• If \(a < 0\), then \(|a| = -a\).

In the problem \(3 \leq |2x - 1|\), the purpose is to find the set of values for \(x\) that makes the expression \(|2x - 1|\) satisfy the inequality. To solve it, you split the absolute value into two cases based on the definition:

• When the expression inside the absolute value, \(2x - 1\), is positive or zero, it directly equals itself: \(2x - 1 \geq 3\).
• When it's negative, the expression equals its negative: \(-(2x - 1) \geq 3\).

Each case must be solved separately to find the solution sets that make the original inequality true.

The Role of Algebra in Solving Inequalities

Algebra is a fundamental tool that helps in the resolution of inequalities. It involves finding out unknown variables by manipulating equations and inequalities using algebraic principles and operations.
In the provided exercise, we utilize algebra to break down absolute value inequalities into conditions that can be solved like regular inequalities. Solving \(2x - 1 \geq 3\) involves straightforward algebra: add 1 to both sides and then divide by 2 to isolate \(x\). This gives us \(x \geq 2\). Similarly, solving \( -(2x - 1) \geq 3\) involves simplifying the inequality to \(-2x + 1 \geq 3\), and after rearranging and simplifying, it results in \(x \leq -1\).
Through algebraic techniques, we consolidate the solution by finding the union of both inequalities. Algebra allows us to see a broader picture and derive a comprehensive solution interval. This interval provides the set of all possible values of \(x\) that satisfy the original absolute value inequality, encompassing both ends of the number line: \(x \leq -1\) or \(x \geq 2\).