Question

Rewrite each of the statements without using a modulus sign: $$ |x|<5 $$

Short answer

Question: Rewrite the inequality $|x| < 5$ without using the modulus sign. Answer: $-5 < x < 5$

Step-by-step solution

Understand the inequality involving the absolute value

The given inequality is $$ |x| < 5 $$ This inequality means that the distance between x and 0 on the number line is less than 5 units.

Convert the absolute value inequality into two inequalities

We can rewrite the inequality in terms of x by noting that x will be in the range: $$ -5 < x < 5 $$ which means x is greater than -5 and less than 5 at the same time.

Combine the two inequalities

We have the rewritten inequality without using the modulus sign: $$ -5 < x < 5 $$

Key concepts

Modulus Sign

The modulus sign, often represented as an upright bar or pair of bars enclosing a number or an algebraic expression, is a mathematical symbol used to indicate the absolute value. In the context of the given exercise, where we look at \(|x|<5\), the modulus sign signifies the non-negative value of \(x\), regardless of whether \(x\) itself is positive or negative.

Understanding the modulus sign is crucial when dealing with inequalities because it influences how we interpret the relationship between numbers. It essentially measures the distance a number is from zero on a number line, without considering its direction. Imagine a point moving along the number line: whether it goes to the left or right of zero, the modulus sign is concerned with how far it travels, not which direction it takes.

Inequality

Inequalities help us compare values to determine if one is larger, smaller, or not equal to another. They are essential tools in mathematics, extending our understanding beyond the certainty of equality to the realms of uncertainty, ranges, and conditions.

In our exercise, the inequality \(|x| < 5\) highlights a range of numbers where \(x\) isn't a single value but can be any number that has an absolute value less than 5. This brings us to a key concept in understanding inequalities of this nature: they often define a span or segment rather than a fixed point. The solution to the exercise, \(-5 < x < 5\), reveals that any number between -5 and 5 (not including -5 and 5 themselves) satisfies the original inequality when stripped of the modulus sign.

Number Line

A number line is a visual representation of numbers laid out sequentially along a straight line, providing a clear way to visualise numbers' relationships. Each point on a number line corresponds to a number, and the position of that point relative to zero (the origin) demonstrates its value.

In tackling absolute value inequalities, the number line essentially becomes a map for finding the range of values that solve the inequality. In the exercise \(|x|<5\), when we apply this concept to a number line, we look for all the points that lie within 5 units of zero, both to the left (negative direction) and to the right (positive direction). The result of the exercise, \(-5 < x < 5\), captures this set of points, emphasizing what is so powerful about the number line: it takes a complex condition like an inequality and simplifies it into an easy-to-see-and-understand selection of points.