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Chapter 1: Problem 100

Solve each equation or inequality. $$ |x| \geq-15 $$

Short Answer

Expert verified
All real numbers satisfy the inequality.

Step by step solution

01

Understand Absolute Value Concept

Absolute value of a number represents its distance from zero on the number line, regardless of the direction. Hence, it is always non-negative. For any value of x, \(|x| \geq 0\).
02

Analyze the Inequality

Given \(|x| \geq -15\), note that the right-hand side (-15) is negative. Absolute values are always non-negative, so any real number will satisfy this inequality.
03

State the Solution

Since all real numbers satisfy \(|x| \geq -15\), the solution is \(x \in \mathbb{R}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value
Absolute value is a key concept in mathematics that determines how far a number is from zero on the number line. It does not take direction into account, which means it doesn't matter if the number is positive or negative. The absolute value of any real number is always non-negative.

For example:
  • The absolute value of 5 is 5.
  • The absolute value of -5 is also 5.
  • The absolute value of 0 is 0.

The mathematical representation for absolute value is \(|x|\). This means that if x is a positive number or zero, \(|x|\) is simply x. If x is negative, \(|x|\) is -x. Understanding this foundation is crucial for solving absolute value inequalities.
Inequality
Inequality is similar to an equation, but instead of showing that two expressions are equal, it shows that one expression is either larger or smaller than another. Inequalities use symbols like \(<\), \(>\), \( \leq \), and \( \geq \).

In this exercise, the inequality is \(|x| \geq -15\). Because absolute values are always non-negative (greater than or equal to 0), this inequality is naturally satisfied. A non-negative number will always be greater than or equal to any negative number. Therefore, every real number is a solution to the inequality.
Real Numbers
Real numbers include all positive numbers, negative numbers, and zero. They encompass all the numbers that can be represented on the number line.

When we solve the inequality \(|x| \geq -15\), we determine that all real numbers satisfy this condition. This is why the final solution can be written as \(x \in \mathbb{R}\), indicating that x can be any real number. Understanding the set of real numbers is fundamental to grasping why such inequalities hold true for all x.

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Most popular questions from this chapter

Solve each problem. How much water must be added to 6 gal of a \(4 \%\) insecticide solution to reduce the concentration to \(3 \% ?\) (Hint: Water is \(0 \%\) insecticide.)

Find the open interval in which \(x\) must lie in order for the given condition to hold. \(y=5 x+12,\) and the difference of \(y\) and 4 is less than 0.0001.

Solve each equation. $$ |2 x-6|=|2 x+11| $$

Solve each investment problem. Mona received a year-end bonus of \(\$ 17,000\) from her company and invested the money in an account paying \(6.5 \%\). How much additional money should she deposit in an account paying \(5 \%\) so that the return on the two investments will be \(6 \% ?\)

Solve each equation or inequality. $$ |0.5 x-3.5|+0.2 \geq 0.6 $$
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