Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-32: Rewrite the expression without using the absolute value symbol, and simplify the result. $$ |7+x| \text { if } x \geq-7 $$

Step-by-Step Solution

Answer

\(|7+x| = 7+x\) for \(x \geq -7\).

Step 1: Understand the Definition of Absolute Value

The absolute value of a number represents its distance from zero on the number line without considering its direction. For any real number \(a\), \(|a| = a\) if \(a \geq 0\), and \(|a| = -a\) if \(a < 0\). In this case, \(|7+x|\) represents the absolute value of \(7+x\).

Step 2: Analyze the Condition Given

The problem states that the condition \(x \geq -7\) must be satisfied. Simplifying \(7 + x \geq 0\) would follow as: \(x \geq -7\). This means that the expression \(7 + x\) is non-negative.

Step 3: Rewrite the Expression without Absolute Value

Since \(7 + x \geq 0\) for \(x \geq -7\), the expression \(|7+x|\) is simply \(7+x\). Therefore, there's no need to change the sign.

Step 4: Simplify if Possible

There is no further simplification needed as the expression \(7+x\) is already in its simplest form without the absolute value symbol.

Key Concepts

Expression Simplification

Simplifying expressions means making them easier to understand or solve by transforming them into an equivalent, usually shorter or more straightforward, form. Here's how we approached simplifying the expression given in the original exercise:

• The expression we began with was \(|7 + x|\).
• Our task was to remove the absolute value signs under the condition that \(x \geq -7\).
• Since \(7 + x\) is non-negative when \(x \geq -7\), the expression simplifies directly to \(7 + x\) without further alterations.

Simplifying expressions often involves reducing them to a form that is easier to interpret, compute, or use in further calculations. This step can involve various techniques, depending on the initial form and the specific conditions provided. In this case, understanding the condition enabled a straightforward simplification.

Real Numbers

Real numbers encompass all the numbers that can be found on the number line. They include both rational and irrational numbers, covering:

• Whole numbers (e.g., 0, 1, 2)
• Integers (e.g., -3, 0, 4)
• Fractions (e.g., 1/2, -3/4)
• Decimals (e.g., 0.75, -2.5)
• Irrational numbers (e.g., \(\sqrt{2}\), \(\pi\))

Real numbers are essential as they provide a way to measure and represent quantities in mathematics and the real world. The exercise condition that \(x \geq -7\) indicates that \(x\) is part of the real number system. The real numbers provide a versatile tool for dealing with broad mathematical concepts, making them highly valuable in simplifying and solving expressions.

Absolute Value Definition

The absolute value of a number is a measure of its distance from zero on the number line. This geometric interpretation simplifies many mathematical expressions and provides a consistent way to handle positive and negative numbers. The definition can be summarized as follows:

• \(|a| = a\) if \(a \geq 0\)
• \(|a| = -a\) if \(a < 0\)

For any expression inside the absolute value, like \(|7 + x|\), understanding this concept allows us to know when and how to remove the absolute value signs based on given conditions. The condition, \(7 + x \geq 0\), enabled us to simplify \(|7 + x|\) directly to \(7 + x\). Absolute value helps in translating conditions and expressions into simpler, actionable forms.