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Chapter 4: Problem 14

Determine the value of each expression. a. \(|-5-3|\) c. \(|2-6|\) b. \(|6-2|\) d. \(-2|-1+3|+|-5|\)

Short Answer

Expert verified
The values are: 8, 4, 4, 1.

Step by step solution

01

Calculate \(|-5-3|\)

First, simplify inside the absolute value: \[-5 - 3 = -8\]. Next, take the absolute value: \[|-8| = 8\].
02

Calculate \(|2-6|\)

First, simplify inside the absolute value: \[2 - 6 = -4\]. Next, take the absolute value: \[|-4| = 4\].
03

Calculate \(|6-2|\)

First, simplify inside the absolute value: \[6 - 2 = 4\]. Next, take the absolute value: \[|4| = 4\].
04

Calculate \-2| -1 + 3 | + |-5|\)

First, simplify inside the absolute value: \[-1 + 3 = 2\], \[|-5| = 5\]. Next, plug back in: \[-2|2| + 5 = -2 \times 2 + 5 = -4 + 5 = 1\].

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Key Concepts

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

Understanding Algebra
Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions. It allows us to generalize arithmetic operations and solve for unknown variables.
Every algebraic expression involves a combination of constants (known values) and variables (unknown quantities). Here are the key elements:
  • Variables: Represented by letters such as \( x \) or \( y \).
  • Constants: Fixed values like 2, -5, or \( \frac{3}{4} \).
  • Operations: Include addition, subtraction, multiplication, and division.
In our exercise, we use these elements to perform operations within absolute value expressions. Ensuring you fully grasp basic algebra properties will help you solve more complex problems effortlessly.
Absolute Value Equations
Absolute value refers to the distance a number is from zero on a number line, regardless of direction. It is always non-negative. The absolute value of a number \( x \) is denoted as \( |x| \). Here are some properties of absolute values:
  • \( |a| \,=\, a \) if \( a \) is positive or zero
  • \( |a| \,=\, -a \) if \( a \) is negative
Let's apply these properties to our exercise steps:
  • Step 1: \( |-5 - 3| \,=\, |-8| \,=\, 8 \)
  • Step 2: \( |2 - 6| \,=\, |-4| \,=\, 4 \)
  • Step 3: \( |6 - 2| \,=\, 4 \)
For Step 4:
  • First, simplify inside the absolute value: \( |-1 + 3| \,=\, |2| \,=\, 2 \) and \( |-5| \,=\, 5 \)
  • Next, substitute back: \( -2|2| + 5 \,=\, -2 \times 2 + 5 \,=\, -4 + 5 \,=\, 1 \)
Knowing how to work with absolute values simplifies finding correct answers in algebraic equations.
Simplifying Expressions
Simplifying expressions means making them easier to comprehend or work with by combining like terms and reducing complexity. Here are some key tips:
  • Combine like terms: Terms that have the same variable raised to the same power. For example, \( 3x + 2x = 5x \).
  • Follow the order of operations: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right) - abbreviated as PEMDAS.
For our example, simplifying inside the absolute value comes first. Once simplified, then apply the absolute value:
  • For \( |-5-3| \), first simplify to \( -8 \), then take absolute value to get \( 8 \)
  • For \( |2-6| \), simplify to \( -4 \) first, then absolute value is \( 4 \)
  • For \( |6-2| \), simplify to \( 4 \), absolute value is \( 4 \)
  • In the final part, simplify \( |-1+3 \rightarrow 2 \) then look at absolute values in parts: \( -2|2| + 5 \,=\, -2 \times 2 + 5 \,=\, 1 \)
Practicing these methods of simplification makes solving algebraic expressions and equations more manageable and accurate.

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Most popular questions from this chapter

a. For any nonzero real number \(a\), what can we say about the sign of the expression \((-a)^{n}\) when \(n\) is an even integer? What can we say about the sign of \((-a)^{n}\) when \(n\) is an odd integer? b. What is the sign of the resulting number if \(a\) is a positive number? If \(a\) is a negative number? Explain your answer.

In the United States, land is measured in acres and one acre is 43,560 sq \(\mathrm{ft}\) a. If you buy a one-acre lot that is in the shape of a square, what would be the length of each side in feet? b. A newspaper advertisement states that all lots in a new housing development will be a minimum of one and a half acres. Assuming the lot is rectangular and has \(150 \mathrm{ft}\) of frontage, how deep will the minimal-size lot be? If the new home owner wants to fence in the lot, how many yards of fencing would be needed? c. The metric unit for measuring land is the square hectometer. (A hectometer is a length of 100 meters.) Find the size of a one-acre lot if it were measured in square hectometers. d. A hectare is 100 acres. How many one-acre lots can fit in a square mile? How many hectares is that?

An equilateral triangle has sides of length \(8 \mathrm{~cm}\). a. Find the height of the triangle. (Hint: Use the Pythagorean theorem on the inside back cover.) b. Find the area \(A\) of the triangle if \(A=\frac{1}{2} b h\).

A TV signal traveling at the speed of light takes about \(8 \cdot 10^{-5}\) second to travel 15 miles. How long would it take the signal to travel a distance of 3000 miles?

The pH scale measures the hydrogen ion concentration in a liquid, which determines whether the substance is acidic or alkaline. A strong acid solution has a hydrogen ion concentration of \(10^{-1}\) M. One \(\mathrm{M}\) equals \(6.02 \cdot 10^{23}\) particles, such as atoms, ions, molecules, etc., per liter, or 1 mole per liter. \(^{6}\) A strong alkali solution has a hydrogen ion concentration of \(10^{-14} \mathrm{M}\). Pure water, with a concentration of \(10^{-7} \mathrm{M},\) is neutral. The \(\mathrm{pH}\) value is the power without the minus sign, so pure water has a \(\mathrm{pH}\) of \(7,\) acidic substances have a pH less than \(7,\) and alkaline substances have a \(\mathrm{pH}\) greater than 7 . a. Tap water has a pH of 5.8 . Before the industrial age, rain water commonly had a pH of about \(5 .\) With the spread of modern industry, rain in the northeastern United States and parts of Europe now has a \(\mathrm{pH}\) of about \(4,\) and in extreme cases the \(\mathrm{pH}\) is about \(2 .\) Lemon juice has a \(\mathrm{pH}\) of 2.1. If acid rain with a pH of 3 is discovered in an area, how much more acidic is it than preindustrial rain? b. Blood has a pH of 7.4 ; wine has a pH of about 3.4. By how many orders of magnitude is wine more acidic than blood?
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