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Chapter 9: Problem 48

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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 \( (-\infty, \infty)\).

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Most popular questions from this chapter

Graph each compound inequality. \(y \leq-x-1\) or \(x \geq 6\)

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Is \((3,5)\) in the solution set of the compound inequality \(x-y \geq-6\) or \(2 x+y<7 ?\) Why or why not?

The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$9 \leq|7-8 q|$$

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