Problem 106 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. 1.85 more than the sum of -1.25 and -4.75

Step-by-Step Solution

Answer

-40

Step 1: Understand the Phrase

First, break down the given phrase: 'The product of 4 and -7, added to -12.' This means you need to find the product of 4 and -7, and then add -12 to the result.

Step 2: Calculate the Product

The product of 4 and -7 can be written as the multiplication: \[4 \times (-7)\]Calculate this to get:\[4 \times (-7) = -28\]

Step 3: Add -12 to the Product

Now, take the result from the previous step, which is -28, and add -12 to it:\[-28 + (-12)\]This can be simplified by combining the negative numbers:\[-28 + (-12) = -28 - 12 = -40\]

Step 4: Write the Final Expression

The numerical expression for the given phrase is:\[4 \times (-7) + (-12)\]And after simplification, it equals:\[-40\]

Key Concepts

product of integers

Multiplying integers means finding their product. To find the product of two integers, you multiply them together. Here, you are dealing with one positive number (4) and one negative number (-7). When you multiply a positive number by a negative number, the result is always negative.
So, the product of 4 and -7 can be written as \(4 \times -7\). Calculating this product, you get \(4 \times -7 = -28\).

Key takeaways:

• Multiplying positive and negative numbers results in a negative number.
• The numerical expression for the product is written as a multiplication.

addition of negative numbers

Adding negative numbers might seem tricky, but it's simpler than you think. When you add a negative number to another number (positive or negative), you move left on the number line, which makes the sum smaller.
In the example given, you add -12 to -28:
\(-28 + (-12)\). Adding -12 to -28 moves further left on the number line, as if you are subtracting 12 more from -28:
\(-28 + (-12) = -40\).

Key points:

• Adding two negative numbers results in a more negative number.
• This process is the same as moving further left on a number line.

expression simplification

Simplifying numerical expressions involves performing arithmetic operations step-by-step. Here, you first find the product of the integers and then deal with the addition of negative numbers.
The problem provides a phrase that translates to a numerical expression: 'The product of 4 and -7, added to -12'. First, calculate \(4 \times -7\) to get -28. Then, add -12 to -28:
\(-28 + (-12) = -40\).
The simplified form of the expression \(4 \times -7 + (-12)\) is -40.

In summary:

• Break down the expression into smaller steps.
• Perform multiplication before addition or subtraction.
• Combine like terms to simplify the expression.