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Chapter 1: Problem 8

Write the given expression without the absolute value symbols. \(|-h|,\) if \(h\) is negative

Short Answer

Expert verified
The expression \\(|-h|\\) simplifies to \\(h\\) when \\(h\\) is negative.

Step by step solution

01

Understand Absolute Value Concept

The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\).
02

Apply the Absolute Value Definition

Since \(h\) is negative, we apply the absolute value definition for negative numbers. Therefore, \(|-h| = -(-h) = h\).
03

Simplify the Expression

Simplifying the expression from Step 2, \(-(-h) = h\), which resolves to \(h\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Negative Numbers
Negative numbers are numbers less than zero. They are represented with a minus sign (-) and appear to the left of zero on the number line. When dealing with negative numbers, it's important to remember:
  • They indicate a position opposite to positive numbers on the number line.
  • Subtracting a negative number is the same as adding its absolute value.
  • Multiplying two negative numbers yields a positive product.
Recognizing the role of negative numbers helps in various mathematical operations, including measuring distances on a number line, and handling temperatures below zero.
Interpreting the Number Line
The number line is a visual representation of numbers in order along a straight line. Each number corresponds to a unique point on the line. Here's what you need to know about number lines:
  • The line extends infinitely in both directions, with numbers becoming progressively larger towards the right and smaller towards the left.
  • Zero marks the central point, separating positive numbers from negative numbers.
  • The absolute value of a number represents how far it is from zero, making it a helpful tool for visualizing absolute distances.
By understanding the number line, you can easily see the relative positions of numbers and better grasp the concept of absolute value.
Simplifying Mathematical Expressions
Simplifying expressions involves reducing them to their most basic form. This process makes it easier to work with the expressions in calculations or equations. Key steps include:
  • Combining like terms, such as adding similar numbers or multiplying variables with the same base.
  • Eliminating unnecessary elements like absolute values when possible.
  • Following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).
Simplifying expressions is crucial for solving algebra problems more effectively and efficiently. It enables you to directly see relationships between variables and constants.

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Most popular questions from this chapter

Use binomial expansion to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(2(h+1)^{3}-5(h+1)^{2}+3\) (b) \(\lim _{h \rightarrow 0} \frac{2(h+1)^{3}-5(h+1)^{2}+3}{h}\)

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=2-\sqrt{x+5}\)

Sketch the semicircle defined by the given equation. \(y=-\sqrt{9-(x-3)^{2}}\)

The given algebraic expression is an unsimplified answer to a calculus problem. Simplify the expression. $$ \frac{2 x(-4 x+6)^{1 / 2}-x^{2}\left(\frac{1}{2}\right)(-4 x+6)^{-1 / 2}(-4)}{\left[(-4 x+6)^{1 / 2}\right]^{2}} $$

Use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). (a) \(\frac{x^{2}-7 x+6}{x-1}\) (b) \(\lim _{x \rightarrow 1} \frac{x^{2}-7 x+6}{x-1}\)
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