Problem 102 - Contemporary Precalculus

Question

Write the given expression without using absolute values. $$\left|-2-y^{2}\right|$$

Step-by-Step Solution

Answer

Question: Write the given expression without using absolute values: $$|-2 - y^2|$$ Answer: For all real numbers \(y\), the given expression can be written without using absolute values as: $$2 + y^2$$

Step 1: Case 1: Expression inside absolute value is non-negative

First, let's analyze the case where the expression inside the absolute value is non-negative: \(-2 - y^2 \ge 0\). In this situation, the absolute value will not change the expression and we simply have: $$|-2 - y^2| = -2 - y^2$$ We also need to find for which values of \(y\) this case is true. To do so, we will solve the inequality: $$-2 - y^2 \ge 0$$ $$y^2 \le -2$$ Since the square of a real number is always non-negative, there is no value of \(y\) for which \(y^2 \le -2\). That means we don't have to consider this case as it doesn't give us any possible values of \(y\).

Step 2: Case 2: Expression inside absolute value is negative

Now, let's analyze the case where the expression inside the absolute value is negative: \(-2 - y^2 < 0\). In this situation, the absolute value will negate the expression to make it non-negative: $$|-2 - y^2| = 2 + y^2$$ We also need to find for which values of \(y\) this case is true. To do so, we will solve the inequality: $$-2 - y^2 < 0$$ $$y^2 > -2$$ Since the square of a real number is always non-negative, this inequality is always true for any \(y \in \mathbb{R}\). Therefore, this case is valid for all real values of \(y\).

Step 3: Final Result

Since we found that Case 2 is true for all values of \(y\), and Case 1 is not true for any values of \(y\), we can write the given expression without using absolute values as: $$\left|-2-y^{2}\right| = 2 + y^{2}$$ This is our final result, valid for all real numbers \(y\).

Key Concepts

Real Numbers

Real numbers are the foundation for many topics in mathematics and they include all the numbers on the number line. This means they encompass:

• Whole numbers like 0, 1, 2, 3...
• Decimals such as 3.14, -0.567...
• Fractions including 1/2, -3/4...
• Irrational numbers like \(\sqrt{2}\) or \(\pi\)

Real numbers can be positive, negative, or zero. They are used in equations, expressions, and inequalities involving concepts like absolute value. In the exercise, real numbers determine whether the expression inside the absolute value sign is positive or negative, which influences how we remove the absolute value sign. Understanding real numbers helps in figuring out domains and the solutions to equations.

Inequalities

Inequalities express a relationship between two expressions showing that one is larger or smaller than the other. They use symbols like:

• \(>\) greater than
• \(<\) less than
• \(\ge\) greater than or equal to
• \(\le\) less than or equal to

In the provided exercise, inequalities help determine when the expression inside the absolute value is negative or non-negative. We solve these inequalities to decide how the absolute value will affect the expression. For example, if we have \(-2 - y^2 < 0\), this tells us that the expression inside the absolute value is negative, allowing us to rewrite it as \(2 + y^2\). Thus, inequalities help us break down the cases needed to simplify expressions effectively.

Expression Simplification

Simplifying expressions is about reducing mathematical expressions to their simplest form while maintaining equivalence. For absolute values, this often involves discussing conditions where an expression is positive or negative. In the problem, simplification involved rewriting \(|-2 - y^2|\) without the absolute value. We determine when the inside content is negative or not and adjust accordingly:

• If \(-2 - y^2\) is non-negative initially (though not applicable to our solution as we found), \(\left|-2-y^2\right| = -2 - y^2\).
• If \(-2 - y^2\) is negative, \(\left|-2-y^2\right| = 2 + y^2\).

Since \(-2 - y^2\) is always negative for real numbers \(y\), the simplification results in \(2 + y^2\). Understanding how to simplify by considering these conditions ensures we handle expressions correctly and clearly.