Problem 39 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ -6+1-7 $$

Step-by-Step Solution

Answer

The result is -12.

Step 1: Solve the First Operation

Start by solving the first operation in the expression, which is to add -6 and 1. Compute -6 + 1 = -5.

Step 2: Solve the Second Operation

Next, take the result from the previous step and subtract 7 from it. Compute -5 - 7 = -12.

Step 3: Verify with a Calculator

To ensure accuracy, input the entire expression into a calculator: -6 + 1 - 7. Confirm that the calculator result is also -12.

Key Concepts

Adding Integers

Understanding how to add integers is fundamental in math. When adding negative and positive numbers, it helps to think in terms of gain and loss. Suppose you have a temperature of -6 degrees, and it increases by 1 degree. You combine negative six and positive one in this context. It's similar to losing six dollars and finding one dollar afterward. Simply put, you first encounter a loss, then a small gain.

• When you add two positive integers, the result is positive.
• When you add two negative integers, the result is negative, just more negative.
• But when you add a negative integer to a positive integer, or vice versa, you must consider the magnitude and direction, which results in the subtraction of absolute values.

In this case, \[-6 + 1 = -5\]. This means that the result is between -6 and 0, specifically moving one step closer to zero.

Subtracting Integers

Moving ahead to subtracting integers, it's akin to comparing two numbers' absolute values where direction matters. To understand subtraction, let's consider it as "adding the opposite." For instance, \[-5 - 7\] can be viewed as adding \[-5\] and the opposite of \[7\], which is \[-7\].

While adding and subtracting negative numbers may seem tricky, think of it as removing more from what you already owe or lack. It's just enhancing the negative value you have, just like converting a cold day colder.

• If you subtract a positive number from a negative one, the result becomes more negative.
• If you subtract a negative number, you essentially add to the number.

Here, \[-5 - 7 = -12\]. Since you started with something negative and made it more negative by adding \[-7\], the result is \[-12\].

Using a Calculator for Verification

Using a calculator is a handy way to verify your results, especially when dealing with integer operations, which can sometimes be error-prone if done mentally. When solving problems like \[-6 + 1 - 7\], plug in each number and operation precisely to avoid mistakes.

Calculators follow the order of operations, just like you should on paper. So ensure your operations are entered exactly as they appear in the expression:

• Start with the equation: enter \[-6\].
• Add \[1\] and press the plus key followed by \[1\].
• Then subtract \[7\] by pressing the minus key followed by \[7\].

Once entered correctly, the calculator shows the result, which should match your manual calculation, \[-12\]. Calculators keep things straightforward and are valuable tools to double-check your work and build confidence in your math skills.