Question

Write an inequality using an absolute value to describe each statement. \(x\) is more than 5 units from 2 .

Short answer

The inequality is \( |x - 2| > 5 \).

Step-by-step solution

Understand the Problem

The statement says that the variable, x, is more than 5 units away from the number 2. We need to translate this into an inequality using absolute value.

Define the Absolute Value Inequality

The absolute value represents the distance between two numbers on the number line. So, \( |x - 2| \) will give us the distance between x and 2.

Set Up the Inequality

Since x is more than 5 units away from 2, the inequality will be \( |x - 2| > 5 \). This means that the distance from x to 2 is greater than 5 units.

Key concepts

absolute value

Absolute value is a fundamental concept in mathematics, representing the distance a number is from zero on a number line. The absolute value of a number is always positive or zero, never negative. For example, the absolute value of both -3 and 3 is 3, written mathematically as \(|-3| = 3\) and \(|3| = 3\). The notation for absolute value involves vertical bars around the number or expression, like this: \(|x|\).

When dealing with absolute value in equations and inequalities, you are essentially measuring distances. This is crucial for understanding the absolute value inequalities, where you might need to determine how far a number is from a specific point, regardless of direction.

inequality

Inequalities are expressions that show the relationship between two values in which one is greater than, less than, greater than or equal to, or less than or equal to the other. The common symbols for inequalities are:

• \(>\) (greater than)
• \(<\) (less than)
• \(≥\) (greater than or equal to)
• \(≤\) (less than or equal to)

In the context of absolute value inequalities, these symbols help us understand if a number lies within a certain range or outside of it. For example, \(|x - 2| > 5\) means that the distance between x and 2 is greater than 5 units.

distance on a number line

Understanding distances on a number line is crucial when working with absolute values. The distance between any two points on a number line is simply the absolute value of the difference between those points. For example, the distance between 3 and 8 is \(|3 - 8| = 5\) or \(|8 - 3| = 5\).

When interpreting the inequality \(|x - 2| > 5\), you can think of it as describing all values of x that are either more than 5 units to the right of 2 or more than 5 units to the left of 2. This means x can be larger than 7 (2+5) or smaller than -3 (2-5). The number line helps visualize how these distances work.

algebra

Algebra involves working with variables and mathematical symbols to solve problems. Solving absolute value inequalities like \(|x - 2| > 5\) involves understanding both the absolute value and inequality concepts and combining these with algebraic skills.

To solve such inequalities, we split them into two separate cases. For \(|x - 2| > 5\), the two cases are: \(x - 2 > 5\) and \(x - 2 < -5\). Solving these cases separately:
1. \(x - 2 > 5\) leads to \(x > 7\)
2. \(x - 2 < -5\) leads to \(x < -3\)

Thus, the solution to the original inequality is that x can be any value greater than 7 or any value less than -3. This showcases how algebra ties in with absolute values and inequalities to find solutions.