Problem 55 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(35+(-14)+(-19)+(-5)\)

Step-by-Step Solution

Answer

-3

Step 1 - Group Positive and Negative Numbers

Separate the positive and negative numbers in the given expression. The expression is: \(35 + (-14) + (-19) + (-5)\)Positive: \(35\)Negative: \(-14, -19, -5\)

Step 2 - Add Negative Numbers

Sum the negative numbers together first: \(-14 + (-19) + (-5) = -14 - 19 - 5\)Calculate: \(-14 - 19 = -33\)\(Then, -33 - 5 = -38\)So, \(-14 + (-19) + (-5) = -38\)

Step 3 - Add the Result to the Positive Number

Add the result of the negative sum to the positive number:\(35 + (-38)\)This is equivalent to:\(35 - 38 = -3\)

Step 4 - Verify Using a Number Line (Optional)

To verify, plot the addition on a number line if desired:Starting at 35, move 14 steps left to 21, then 19 steps left to 2, and 5 steps left to -3. This confirms the result.

Key Concepts

Grouping Positive and Negative Numbers

When faced with a problem involving both positive and negative numbers, it is helpful to separate them first. This makes it easier to handle the arithmetic. Let's look at our example: \(35 + (-14) + (-19) + (-5)\).
Here, we have:
Positives: \(35\)
Negatives: \(-14, -19, -5\).
Splitting them up like this lets you deal with simpler sums first.

Addition of Negative Numbers

When adding negative numbers, think of it as subtracting their absolute values.
For example, adding \(-14 + (-19) + (-5)\) is the same as subtracting those numbers from zero. This makes it:

• \(-14 + (-19) = -33\)
• Then, \(-33 + (-5) = -38\)

So, the sum of our negative numbers is \(-38\).
By handling them separately, you avoid the hassle of mixing positive and negative integers initially.

Checking Answers Using Number Line

To verify our answer, a number line is a very useful tool.
Let's check the final sum \(35 + (-38)\):

• Start at 35.
• Move 14 steps left to land on 21.
• Move 19 more steps left to land on 2.
• Finally, move 5 steps left to land on \(-3\).

We see that the end position is \(-3\), which confirms our answer.

Using the number line can often make it clearer why the addition works this way.