Problem 41 - Algebra and Trigonometry

Question

\(23-48\) Solve the inequality. Express the answer using interval notation. $$ 8-|2 x-1| \geq 6 $$

Step-by-Step Solution

Answer

\([-\frac{1}{2}, \frac{3}{2}].\)

Step 1: Understand the Inequality

The inequality can be written as \[ 8 - |2x - 1| \geq 6. \]To solve it, we first need to isolate the absolute value expression.

Step 2: Isolate the Absolute Value

Subtract 8 from both sides of the inequality to isolate the absolute value term:\[ 8 - |2x - 1| - 8 \geq 6 - 8 \]This simplifies to \[-|2x - 1| \geq -2. \]

Step 3: Remove the Negative Sign

Multiply both sides of the inequality by -1 to remove the negative sign, remembering to reverse the inequality:\[ |2x - 1| \leq 2. \]

Step 4: Set Up Two Inequalities

Since we have an absolute value inequality, we split it into two separate inequalities:1. \(2x - 1 \leq 2\)2. \(2x - 1 \geq -2\)

Step 5: Solve the First Inequality

Solve the inequality \(2x - 1 \leq 2\) by adding 1 to both sides:\[ 2x \leq 3 \]Then, divide by 2:\[ x \leq \frac{3}{2}. \]

Step 6: Solve the Second Inequality

Solve the inequality \(2x - 1 \geq -2\) by adding 1 to both sides:\[ 2x \geq -1 \]Then, divide by 2:\[ x \geq -\frac{1}{2}. \]

Step 7: Combine the Solutions

Combine the solutions of the two inequalities to express the solution in interval notation:\[ -\frac{1}{2} \leq x \leq \frac{3}{2}. \]

Step 8: Express in Interval Notation

The solution in interval notation is \([-\frac{1}{2}, \frac{3}{2}].\)

Key Concepts

Absolute Value

The absolute value refers to the non-negative value of a number. It describes how far a number is from zero on the number line, without considering direction. In mathematical terms, the absolute value of a number \(a\) is denoted as \(|a|\). For example:

• The absolute value of 5 is \(|5| = 5\).
• The absolute value of -5 is \(|-5| = 5\).

This concept is crucial in solving absolute value inequalities like \(|2x - 1| \leq 2\). Here, the goal is to find a range of \(x\) values that make the expression inside the absolute value not exceed 2 in magnitude.
By breaking down the inequality into two cases, we establish two separate scenarios to solve:

• One where the expression inside is less than or equal to 2.
• Another where it is greater than or equal to -2.

This dual nature of the absolute value is what allows us to solve for \(x\) effectively.

Interval Notation

Interval notation is a shorthand used in mathematics to describe the set of solutions to an inequality. For example, an interval notation such as \([-\frac{1}{2}, \frac{3}{2}]\) indicates all the values between \(-\frac{1}{2}\) and \(\frac{3}{2}\), inclusive.
Each bracket has a specific meaning:

• A square bracket "\([\)" means that the endpoint value is included in the interval (closed interval).
• A parenthesis "\(()\)" means that the endpoint value is not included (open interval).

In solving inequalities, converting the solution into interval notation summarizes the solution set efficiently. For the inequality \(-\frac{1}{2} \leq x \leq \frac{3}{2}\), the interval notation is \([-\frac{1}{2}, \frac{3}{2}]\), clearly showing the range of allowable \(x\) values.
This notation is widely used because it is concise and instantly conveys the extent of possible solutions.

Inequality Solution

Solving inequalities involves finding a range of values that satisfy the given statement. In the context of \(8 - |2x - 1| \geq 6\), the process involves several steps.
Initially, isolate the absolute value expression to easier manipulate it, resulting in \(|2x - 1| \leq 2\).
Then understand that to solve \(|A| \leq B\), it translates to solving two inequalities:

• \(A \leq B\)
• \(A \geq -B\)

This ensures finding all possible values of \(x\) that when plugged back, keeps the statement true. By simplifying these inequalities, you end up with:

• \(x \leq \frac{3}{2}\)
• \(x \geq -\frac{1}{2}\)

Combining such results provides a clear depiction of the solution set in interval notation. Managing inequalities also involves checking your solution against the original inequality to verify its correctness.
These steps ensure a comprehensive approach to tackling inequalities, yielding an easy-to-visualize solution range.