Question

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

Short answer

Q: Rewrite the inequality without using the modulus sign: $$|x| \leq 7.5$$. A: The inequality without the modulus sign is $$-7.5 \leq x \leq 7.5$$.

Step-by-step solution

Case 1: x is non-negative (x ≥ 0)

If x is non-negative, then the absolute value of x is just x itself. Therefore, we can rewrite the inequality as: $$ x \leq 7.5 $$ This inequality holds true when x is greater than or equal to 0.

Case 2: x is negative (x < 0)

If x is negative, then the absolute value of x is the negative of x, denoted as -x. Therefore, we can rewrite the inequality as: $$ -x \leq 7.5 $$ Multiplying both sides by -1 (and flipping the inequality sign) gives: $$ x \geq -7.5 $$ This inequality holds true when x is less than 0.

Combining both cases

Finally, we combine both cases to obtain the inequality without the modulus sign: $$ -7.5 \leq x \leq 7.5 $$ This means that x can be any real number between -7.5 and 7.5, inclusive.

Key concepts

Modulus Sign

Understanding the modulus sign, often referred to as the absolute value sign, is crucial when solving mathematical problems involving the distance from zero on a number line. The absolute value of a number, denoted by two vertical bars as in \( |x| \), represents the number's magnitude regardless of its sign.

For example, the statement \( |x| \) essentially asks: how far is the number 'x' from zero? The key takeaway is that whether 'x' is positive or negative, the modulus sign will return a non-negative result. So, \( |3| = 3 \) and \( |-3| = 3 \) because both 3 and -3 are three units away from zero. When dealing with modulus in inequalities, it signifies that we are considering all values of 'x' that lie within a certain distance from zero.

Inequalities

Inequalities are mathematical expressions that communicate how one quantity is less than or greater than another. The language of inequalities extends to symbols such as \( < \), \( > \), \( \leq \), and \( \geq \), indicating 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to', respectively.

In the context of absolute values, stating an inequality like \( |x| \leq 7.5 \) means that the distance of 'x' from zero should not exceed 7.5 units. This encompasses all possible values 'x' can have within that range. Inequalities can appear in various forms and require different approaches depending on whether we are dealing with linear inequalities, polynomial inequalities, or those involving absolute values.

Absolute Value Algebra

Absolute value algebra involves manipulating equations and inequalities that contain the absolute value sign. When isolating variables within such expressions, it's key to consider both the positive and negative scenarios because the absolute value deals with distance, disregarding direction.

Breaking down the given exercise, we view the inequality \( |x| \leq 7.5 \) through two lenses: one where 'x' is non-negative and one where 'x' is negative. This bifurcated approach ensures that we account for all values that 'x' could take to satisfy the original absolute value inequality. Hence, we arrive at a compound inequality \( -7.5 \leq x \leq 7.5 \), which clearly illustrates the range of solutions. This approach is a staple in solving absolute value inequalities and forms the foundation for more complex problem-solving in calculus and real analysis.