Question

Rewrite each of the statements without using a modulus sign: $$ |x|>7.5 $$

Short answer

Question: Rewrite the inequality |x| > 7.5 without using the modulus sign. Answer: x < -7.5 or x > 7.5

Step-by-step solution

Case 1: x is positive

If x is positive, then we can simply remove the absolute value sign and rewrite the inequality as: $$ x > 7.5 $$

Case 2: x is negative

If x is negative, then the absolute value of x represents the negation of x. So, |-x| > 7.5. In this case, we have: $$ -x > 7.5 $$ Multiplying both sides by -1 (and remembering to switch the inequality sign when multiplying by a negative number) gives: $$ x < -7.5 $$

Combining the two cases

Now that we have the inequalities for each case, we can combine them to reflect the full range of x values that satisfy the original inequality: $$ x < -7.5 \quad \text{or} \quad x > 7.5 $$

Key concepts

Inequalities

At the heart of mathematics, inequalities are expressions that compare two values showing that one is either less than, greater than, or unequal to another. In the given exercise, we delve into a specific type of inequality involving absolute values. Inequalities involving absolute values describe a range of values that a variable can take. They are crucial in determining the set of possible solutions rather than just one answer like equations.

When dealing with inequalities, remember these key rules:

• Whenever you multiply or divide by a negative number, the direction of the inequality sign must be flipped.
• Always consider different "cases" since they often indicate how the expression behaves under different conditions (positive or negative values).

Understanding these fundamentals helps in effectively solving inequalities. When combined with absolute values, the approach requires additional steps to ensure accuracy.

Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. It's always non-negative. The symbol for absolute value is vertical bars \(|x|\). In simpler terms, \(|x| = x\) if \(x\) is positive or zero, and \(|x| = -x\) if \(x\) is negative. This means absolute values transform all numbers to their non-negative counterparts.

In the exercise, we are asked to solve an inequality that involves an absolute value \(|x| > 7.5\). To remove the absolute value, we consider two scenarios corresponding to the definition of absolute value:

• For \(x > 0\), the expression simplifies to \(x > 7.5\).
• For \(x < 0\), we rewrite \(|x|\) as \(-x\) leading to \(-x > 7.5\). Solving gives \(x < -7.5\).

By addressing these two cases separately, we find the comprehensive solution for the inequality.

Number Line Representation

Visualizing solutions on a number line enhances understanding by making abstract concepts more concrete. When solving inequalities, it's especially helpful to see the range of values that satisfy the condition.

For the inequality \(|x| > 7.5\), the solution consists of two parts: \(x > 7.5\) or \(x < -7.5\). Plotted on a number line, these solutions create two distinct segments.

• The first segment starts just beyond 7.5 and extends to positive infinity, representing values greater than 7.5.
• The second segment starts just less than -7.5 and extends to negative infinity, showing values less than -7.5.

Using a number line, it's clear there is a "gap" between -7.5 and 7.5 where no solutions exist. This visualization clearly shows the boundaries and ranges of potential solutions, emphasizing how inequalities differ from simple equations, which typically yield discrete solutions.