Problem 48 - Beginning and Intermediate Algebra

Question

Explain why the solution to \(|7 y-3| \geq 0\) is \((-\infty, \infty)\).

Step-by-Step Solution

Answer

The solution to \(|7y-3| \geq 0\) is \((-\infty, \infty)\) because absolute values are always non-negative. Therefore, for any value of y, the expression \(|7y-3|\) will always be greater than or equal to 0, which includes all real numbers.

Step 1: Understanding Absolute Values

An absolute value of a number or expression is the distance of that number or expression from 0 on the number line. Mathematically, it is denoted as \(|x|\), and defined as: \( |x| = \begin{cases} x, & \text{if } x\geq0 \\ -x, & \text{if } x<0 \end{cases}\) The important thing to remember is that absolute values are always non-negative.

Step 2: Analyze the given inequality

The inequality we are given is \( |7y-3| \geq 0\). Since the absolute value of any expression is always non-negative, we can say that for any value of y, the expression \( |7y-3| \) will always be greater than or equal to 0. Therefore, the solution to this inequality includes all real numbers.

Step 3: Write the solution in interval notation

Since the solution includes all real numbers, we represent it using interval notation as \((-\infty, \infty)\). This means that all real numbers from negative infinity to positive infinity are included in the solution. So, the solution to \( |7y-3| \geq 0\) is \( (-\infty, \infty)\).