Problem 40 - Algebra 1

Question

Evaluate the expression. $$ 14+(-7)-12 $$

Step-by-Step Solution

Answer

-5

Step 1: Identify the operations

In the expression \( 14+(-7)-12 \), we have two addition operations and one subtraction operation. The subtraction is implicit in the term '-7', which can be written as '+(-7)'.

Step 2: Perform the first addition operation

First, perform the operation 14 + (-7). Adding a negative number is equivalent to subtracting its absolute value, so 14 + (-7) = 14 - 7 = 7.

Step 3: Perform the second addition operation

Next, perform the operation 7 - 12. Once again, adding a negative number is equivalent to subtracting its absolute value, so 7 - 12 results in a negative number '-5'.

Key Concepts

Addition and Subtraction of Integers

When working with integers, addition and subtraction can seem tricky, but with a little understanding, they become easy to handle. Integers are whole numbers that can be positive, negative, or zero. Adding and subtracting these numbers is essential in various mathematical problems.

Consider the expression given in the original exercise: \( 14 + (-7) - 12 \). To solve this, we need to break down the expression into simpler parts:

• Begin with \( 14 + (-7) \). Here, you are essentially subtracting 7 from 14, which gives you the result of 7.
• Next, handle the subtraction in the expression \( 7 - 12 \). This part involves subtracting 12 from 7. The answer here is -5, indicating a shift into negative territory since 12 is larger than 7.

Solutions become straightforward when you consistently apply these steps with the rules of integer arithmetic. Breaking down the operation makes it a lot more manageable.

Negative Numbers

Negative numbers represent values less than zero. They are a vital part of integers. When dealing with operations that involve negative numbers, it is essential to understand the effect of these numbers on calculations.

For instance, in the given exercise, you encounter negative numbers. One of them appears in the form of \(-7\). When adding \(-7\) to 14, it effectively means you are moving 7 steps down from 14 on a number line. This is equivalent to subtraction.

Further in the expression \( 7 - 12 \), the result is -5 because subtracting a larger number from a smaller one pushes the answer into negative territory. Understanding how negatives work helps you navigate arithmetic expressions smoothly. They are not just numbers with a minus sign; they have a significant impact on the calculations.

Absolute Value

Absolute value is an essential concept in understanding how to perform operations with integers. It is the non-negative value of a number without regard to its sign. For any integer \( a \), the absolute value is represented as \( |a| \).

In the expressions \( 14 + (-7) \) and \( 7 - 12 \) from the problem, the absolute value helps make sense of the operations involving subtraction of negative numbers. \(-7\) has an absolute value of 7, meaning when you add \(-7\) to 14, it’s like taking away 7.

The absolute value is crucial as it allows you to assess the 'size' of a number regardless of its direction on the number line. For instance, both -5 and 5 have an absolute value of 5, signifying that they are equidistant from zero. This concept ensures that calculations can transition seamlessly through positive and negative numbers without misunderstanding their magnitude.