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Chapter 1: Problem 116

Write each sentence as an equation, using \(x\) as the variable. Then find the solution from the set of integers between \(-12\) and \(12,\) inclusive. See Example \(9 .\) 7 less than a number is 2

Short Answer

Expert verified
The number is 9, and it lies within the given range.

Step by step solution

01

Understanding the Sentence

The sentence '7 less than a number is 2' means that if 7 is subtracted from a number, the result is 2.
02

Formulate the Equation

Let the number be represented by the variable \( x \). The equation can be written as: \( x - 7 = 2 \)
03

Solve for x

To isolate \( x \), add 7 to both sides of the equation: \( x - 7 + 7 = 2 + 7 \) which simplifies to \( x = 9 \).
04

Verify the Solution

Check if 9 is within the range of integers between -12 and 12, inclusive. Since 9 lies within this range, it is a valid solution.

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Key Concepts

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

equation formulation
Understanding how to formulate an equation from a word problem is crucial in algebra. In the given exercise, the sentence '7 less than a number is 2' must be translated into mathematical language. The phrase 'a number' suggests a variable, which we denote as \( x \). The phrase '7 less than' means we subtract 7 from \( x \). The word 'is' translates to an equals sign (\( = \)). Thus, the sentence becomes the equation: \( x - 7 = 2 \). This step transforms the problem into a solvable algebraic expression.
integer range
Once we have our equation, we need to find solutions within a specified range鈥攊n this case, integers between \( -12 \) and \( 12 \) inclusive. Integers are whole numbers that can be positive, negative, or zero. The term 'inclusive' means that the boundaries (\( -12 \) and \( 12 \)) are included in the set. Therefore, to solve the equation \( x - 7 = 2 \), we need to find an integer \( x \) that lies within this range. After we find the value of \( x \), we check whether it falls between \( -12 \) and \( 12 \).
variable isolation
Variable isolation involves solving for the variable \( x \) in an equation. In our problem, we start with the equation \( x - 7 = 2 \). The goal is to get \( x \) by itself on one side of the equation. To do this, we perform the inverse operation of subtraction, which is addition. By adding 7 to both sides of the equation, we eliminate the -7 on the left: \( x - 7 + 7 = 2 + 7 \). This simplifies to \( x = 9 \). Now, \( x \) is isolated, and we have found that \( x \) must equal 9.
verification of solution
Verifying the solution ensures that our answer is correct. First, we found \( x = 9 \) as the solution. We must confirm two things: it satisfies the original equation and lies within the specified range. Plugging 9 back into the equation, we get \( 9 - 7 = 2 \), which is true. Next, we check if 9 is between \( -12 \) and \( 12 \). Since it is, our solution is verified. Thus, \( x = 9 \) is the correct and valid solution to our problem.

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