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Chapter 14: Problem 110

For what value of \(x\) will the arithmetic mean of \(x+4\) and \(x+8\) be \(5 ?\)

Short Answer

Expert verified
The value of \(x\) is \(-1\).

Step by step solution

01

Understand the Arithmetic Mean

The arithmetic mean (average) of two numbers, say \(a\) and \(b\), is given by the formula: \( \frac{a + b}{2} \). In this problem, the numbers are \(x+4\) and \(x+8\).
02

Set Up the Equation

We want the arithmetic mean of \(x+4\) and \(x+8\) to be 5. So, we set up the equation: \( \frac{(x + 4) + (x + 8)}{2} = 5 \).
03

Simplify the Equation

Combine like terms in the numerator: \( x + 4 + x + 8 = 2x + 12 \). The equation becomes \( \frac{2x + 12}{2} = 5 \).
04

Solve for \(x\)

To eliminate the fraction, multiply both sides by 2: \(2x + 12 = 10\). Then, solve for \(x\) by first subtracting 12 from both sides: \(2x = -2\). Finally, divide by 2: \(x = -1\).

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Key Concepts

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

Understanding Algebra and Arithmetic Mean
Algebra is a branch of mathematics that uses symbols to represent numbers and quantities in formulas and equations. One important concept is the arithmetic mean, which is the average of a set of numbers. To find the arithmetic mean of two numbers, you add them together and then divide by two. In this exercise, we work with expressions involving algebraic terms, such as \(x+4\) and \(x+8\), which are the numbers whose mean we need.
  • Algebraic Expression: This is a combination of numbers, variables, and arithmetic operations. For example, \(x+4\) and \(x+8\) are algebraic expressions.
  • Arithmetic Mean Formula: For two numbers, \(a\) and \(b\), the mean is \(\frac{a + b}{2}\).
Algebra helps us to set up equations and solve for unknown variables, using the given formulas and relationships.
Forming and Solving Equations
An equation is a mathematical statement indicating that two expressions are equal. In this problem, we form an equation with the arithmetic mean and solve it to find the variable \(x\).
  • Equation Formation: Given the expressions \(x+4\) and \(x+8\), their arithmetic mean is expressed as \(\frac{(x + 4) + (x + 8)}{2}\). We equate this to 5, the desired mean.
  • Simplifying Equations: Combining like terms in the numerator simplifies our expression to \(2x + 12\), resulting in \(\frac{2x + 12}{2} = 5\).
By simplifying the equation, we make it easier to solve for \(x\). This illustrates how algebra is used to transform expressions into a simpler form for easier problem-solving.
Solving for x
The goal in algebra is often to solve for a variable, in this case, \(x\). This requires isolating \(x\) using algebraic operations.
  • Eliminating Fractions: Multiply both sides by 2 to remove the fraction: \(2x + 12 = 10\).
  • Solving for \(x\): Start by subtracting 12 from both sides: \(2x = -2\). Then, divide each side by 2 to isolate \(x\): \(x = -1\).
Each step in solving the equation involves performing operations that maintain the equation's balance. By systematically simplifying and solving, we find the value of \(x\), which is the unknown in our initial problem. This process exemplifies the critical thinking and methodical approach required in algebra.

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Most popular questions from this chapter

Find the first term of the geometric sequence with a common ratio \(-3\) and an eighth term \(-81\).

Find the sum of the first five terms of the geometric sequence if its first term is 5 and the common ratio is \(-6\).

Find the common difference of the arithmetic sequence with a first term of \(-164\) if its 36th term is \(-24.\)

For each arithmetic sequence, find the sum of the specified number of terms. The first 25 terms of \(2,-3,-8, \ldots\)

Find the 15th term of the arithmetic sequence: \(\frac{1}{2}, \frac{1}{4}, 0, \ldots\)
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