Problem 95 - Introductory Algebra for College Students

Question

We know that \(|x|\) represents the distance from 0 to \(x\) on a number line. Use each sentence to describe all possible locations of \(x\) on a number line. Then rewrite the given sentence as an inequality involving \(|x|\) The distance from 0 to \(x\) on a number line is less than 2 .

Step-by-Step Solution

Answer

The statement 'The distance from 0 to \(x\) on a number line is less than 2 ' describes all numbers \(x\) that are within 2 units of 0 on the number line. The inequality that represents this statement is \(|x| < 2\), which can also be expressed as \( -2 < x < 2\).

Step 1: Understanding the Problem

The statement says 'The distance from 0 to \(x\) on a number line is less than 2 '. This means that we're looking for all numbers \(x\) that are within a distance of 2 from 0. This includes all numbers between -2 and 2.

Step 2: Translating to Inequality

The statement can be translated into an inequality involving the absolute value \(|x|\). This is because \(|x|\) exactly describes the distance from 0 to \(x\) on a number line. The statement 'The distance from 0 to \(x\) on a number line is less than 2 ' translates to \(|x| < 2\).

Step 3: Expanding the Absolute Value

We can express the inequality \(|x| < 2\) without the absolute value by considering the definition of the absolute value. The expression \(|x| < 2\) is equivalent to \( -2 < x < 2\). This is because any number \(x\) that's within 2 units from 0, will be somewhere in between -2 and 2 on the number line.

Key Concepts

Distance on a Number Line

Visualizing the distance from a number to zero is crucial in understanding the concept of absolute value. The absolute value of a number, denoted as \(|x|\), measures how far the number is from zero on a number line, regardless of direction. This means that both -3 and 3, for example, are three units away from zero, so both have an absolute value of 3. In this context, when we say "the distance from 0", we are referring to the length of this distance without concern for whether it lies to the left or right of zero.
For instance, consider a scenario where the distance from 0 to \(x\) is less than 2. We are tasked with identifying all points on the number line that lie within 2 units from zero. This would naturally include points between -2 and 2.

Translating Inequalities

When faced with a verbal statement that involves distances, such as "The distance from 0 to \(x\) is less than 2", translating this into a mathematical inequality is key. Translating problems into inequalities involves symbolizing the situation in a mathematical expression. Given that distance is often represented using absolute value for its non-negative property, we turn the statement into \(|x| < 2\). This transformation is akin to converting words into numbers and signs that express the same relationship.
Subsequently, it helps in evaluating or solving further, providing clarity and direction to your problem-solving approach.

Algebraic Expressions

The step from verbal descriptions to mathematical language often involves expressing constraints with algebraic expressions and inequalities. Here, the inequality form \(|x| < 2\) encapsulates the condition described by the problem.
What makes it algebraic is that it involves an operation (absolute value) along with a variable (\(x\)). This shows the power of using algebraic expressions to succinctly communicate complex ideas — such as ranges or zones on a number line — through simple statement forms.

• Absolute value, representing distance, can simplify broader problems by narrowing them into solvable expressions.
• The algebraic expression aids in determining all values of \(x\) fulfilling the given condition.

Inequalities on Number Lines

Once the inequality \(|x| < 2\) is established, expressing this condition on a number line becomes straightforward. On a number line, this inequality translates to shading the region between -2 and 2, excluding the endpoints.
When encountering inequalities, a number line serves as a visual representation that enables clearer comprehension by highlighting the range of possible solutions. This visual aid fortifies understanding:

• It shows clearly that values of \(x\) can be anything from just above -2 to just below 2.
• It visually explains the concept of inclusion and exclusion regarding endpoints in inequalities (open circles or dashed lines for \(<\) or \(>\)).

Utilizing number lines in this way simplifies abstract concepts into visually digestible formats.