Problem 133 - Beginning and Intermediate Algebra

Question

Write each of the following as an expression and evaluate. Twice the sum of \(-5\) and \(-3\)

Step-by-Step Solution

Answer

The expression is \(2 \times (-5 + (-3))\) and evaluates to -16.

Step 1: Understand the Problem

We need to form an expression for 'twice the sum of -5 and -3'. This means we have to find the sum of -5 and -3, and then multiply that result by 2.

Step 2: Calculate the Sum

First, calculate the sum of -5 and -3.\(-5 + (-3) = -8\)

Step 3: Multiply by Two

Take the sum obtained in Step 2, which is -8, and multiply it by 2. \(-8 \times 2 = -16\)

Step 4: Write the Final Expression

The expression that represents 'twice the sum of -5 and -3' is \(2 \times (-5 + (-3))\). After performing the calculations, we find the result to be -16.

Key Concepts

Basic Arithmetic

Arithmetic is the foundation of mathematics that involves performing operations like addition, subtraction, multiplication, and division. In this exercise, we focus on addition and multiplication. Let's start with the addition part.

When you read "the sum of -5 and -3," it refers to adding these two numbers together. In simple terms, we are combining

• -5 units
• -3 units

into one total. If you combine them, you end up with -8, since

• -5 + (-3) equals -8.

This sum then becomes part of the expression that needs to be further adjusted by multiplication.

Arithmetic steps require you to pay close attention to each operation needed to find the correct result.

Negative Numbers

Negative numbers are numbers that are less than zero. They are crucial in arithmetic operations because they often indicate a loss or decrease. For example, -5 can be thought of as going 5 units backward from zero. Handle them with care, as they follow specific rules during arithmetic operations.

When combining negative numbers, such as in this task, the sum becomes more negative. In the example, the sum of -5 and -3 results in -8. This is because we are adding two quantities that both decrease the total, like deepening a hole. When multiplying with negative numbers, the rule of signs means that multiplying two negatives results in a positive. However, a negative times a positive remains negative, which is the case when calculating -8 times 2 in this exercise.

Understanding these rules helps you better solve and simplify expressions involving negative numbers.

Multiplication

Multiplication is repeated addition. For example, multiplying a number by 2 means adding that number to itself. In the exercise, we need to multiply -8 by 2 after finding the sum.

• -8 times 2 refers to adding -8 to itself once, which results in -16.

Here's the key with multiplication: remember the rule of signs mentioned earlier.

• A negative times a positive number results in a negative product.
• A negative times another negative becomes a positive.

The multiplication component is crucial in shaping the final outcome of the expression. It involves both knowing the basic multiplication facts and understanding how signs affect the product. Carefully follow multiplication sequences to get precise results in algebraic expressions and everyday calculations.