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Chapter 8: Problem 120

Find an equivalent inequality with absolute value. \(x\) is less than 1 unit from 5

Short Answer

Expert verified
The equivalent inequality is \( |x - 5| < 1 \).

Step by step solution

01

Understand the problem

We need to find an equivalent inequality using absolute value to express that the distance between the variable x and 5 is less than 1 unit.
02

Translate the distance to an absolute value inequality

The distance between x and 5 being less than 1 can be written as \( |x - 5| < 1 \). The absolute value represents the distance regardless of direction (positive or negative).
03

Verify the inequality

To ensure correctness, interpret \( |x - 5| < 1 \) as being equivalent to \( -1 < x - 5 < 1 \). Adding 5 to all parts of the inequality \( -1 + 5 < x < 1 + 5 \) should help confirm the result: \( 4 < x < 6 \).

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Key Concepts

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

distance
Distance in mathematics refers to the space between two points on a number line or in a coordinate system. When we say that a value is less than 1 unit from another value, we are talking about the gap between these two points.
In the problem, we need to express that the variable \( x \) is less than 1 unit away from the number 5.
This can be visualized as the space between \( x \) and 5 on a number line being smaller than 1 unit.
Because distance is always positive, regardless of direction, absolute value comes into play to represent this concept.
absolute value
Absolute value is a mathematical function that measures the distance of a number from zero, regardless of its direction on the number line. It is denoted by vertical bars, like this: \( |x| \).
In this problem, we're asked to show that \( x \) is less than 1 unit from 5, which involves using absolute value.
We write this as \( |x - 5| < 1 \).
The expression \( |x-5| \) represents the distance between \( x \) and 5.
This means that no matter if \( x \) is to the left or right of 5 on the number line, the gap is less than 1 unit.
Understanding absolute value will help you grasp why the inequality represents this distance without considering the direction.
inequality
An inequality shows the relationship between two values where one is not necessarily equal to the other. It uses symbols like \(<\), \(>\), \(\leq\), and \(\geq\).
In the given exercise, the inequality used is \( < \), indicating that one value is less than another.
So, when we say \( |x - 5| < 1 \), it tells us that the distance between \( x \) and 5 is less than 1 unit.
To further understand this, we can break down the inequality into two separate linear inequalities: \( -1 < x - 5 < 1 \).
By solving this, we get the range \( 4 < x < 6 \).
Understanding inequalities helps us place bounds and limit the values of variables.
linear equations
Linear equations are algebraic equations where the highest power of the variable is 1. They form straight lines when graphed.
In this exercise, the absolute value inequality can be translated into linear inequalities.
First, starting from \( |x - 5| < 1 \), we break that into \( -1 < x - 5 < 1 \).
Next, we solve these linear inequalities separately.
Adding 5 to all parts of the inequality gives us two linear inequalities: \( 4 < x < 6 \).
Linear equations like these help us define a range for the variable \( x \) and simplify problem-solving.
Understanding how to manipulate and solve linear equations is crucial for working with more complex inequalities.

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Most popular questions from this chapter

Solve by graphing. $$ x-1=-5 $$

Classify each of the following statements as either true or false. To solve \(\frac{x-5}{x+4} \geq 0\) using intervals, we divide the number line into the intervals \((-\infty,-4),(-4,5)\) and \((5, \infty)\).

Solve. $$ 2 \leq|x-1| \leq 5 $$

Find an equivalent inequality with absolute value. \(x\) is less than 2 units from 7 .

Find the domain of each function. $$ f(x)=\sqrt{x^{2}+8 x} $$
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