Subtracting Integers

Subtracting Integers exercise

• Find the integer that brings the seesaw back into balance.

  Hints

  Which of the numbers on the left side of the seesaw do not have an opposite pair on the right side?

  The absolute value of the left side is $16$. Remember, absolute value is the value of a number without the sign.

  Solution

  If we look at the left side of the equation, we see the integers $-7$ and $-9$. On the right-hand side, we have a $+7$.

  The opposite of $-7$ is $+7$, since the absolute value of these numbers is equal.

  Since there's also a $-9$ on the left-hand side, we can balance the seesaw by putting a number that is equal to the opposite of $-9$ on the right-hand side.

  In this case, a $+9$ would balance the equation (seesaw).

• Describe how to solve the equation $-2 - (-4) -3$.

  Hints

  Subtracting a negative number is the same as adding a positive number.

  Subtracting a positive number is the same as adding a negative number.

  Remember to use the proper Order of Operations. (PEMDAS)

  Solution

  Let's start with the given equation:

  • $(-2)-(-4)-3$

  First, we can simplify the signs. Knowing that subtracting a negative number is the same as adding a positive number, we can simplify this equation as follows:

  • $(-2)+(+4)-3$

  • $(-2)+4-3$

  At this point, we can solve. Alternatively, if we need more support, we can change the next minus sign to a plus sign and rewrite the equation like so:

  • $(-2)+4+(-3)$

  Now that we have our equation, we just need to follow the order of operations to solve. Remember, we have to work left to right.

  • $(-2)+4=2$

  • $2+(-3)=-1$

  The answer is $-1$!

• Calculate the temperature change by subtracting integers.

  Hints

  Subtracting a negative number is the same as adding a positive number.

  To calculate the difference of two numbers they have to be subtracted.

  The result is always a positive value.

  Solution

  The temperature on day one is $4$ degrees. The temperature on day two is $-3$ degrees. In order to find the difference in the temperatures, you should subtract $-3$ from $4$. We can write this equation as follows:

  $4-(-3)$

  Since we know that subtracting a negative number is the same as adding a positive number, we can rewrite the equation and solve:

  $4+3=7$

  So the temperature changed by $7$ degrees.

  $\begin{array}{rcl} 4 - (-3) & = & 4 + 3\\ & = & 7 \end{array}$

• Evaluate John's final score by adding and subtracting integers.

  Hints

  Adding a negative number is the same as subtracting a positive number.

  Subtracting a negative number is the same as adding a positive number.

  To solve mathematical expressions, follow the rules of PEMDAS.

  Solution

  Let's look at John's throws:

  John's first throw results in a positive $2$. On John's second throw, he hits $(+/-)$ and a $3$ on his third throw. The fourth throw is a $(-/-)$, and he hits a $6$ on his final throw.

  That will leave us with $2+(-3)-(-6)$. Using the Order of Operations (PEMDAS), we can now solve from left to right.

  $\begin{array}{rcl} 2 + (-3) -(-6) & = & ~~~2 + (-3) + 6\\ & = & -1 + 6 \\ & = & ~~~5 \end{array}$

  John's final score is $5$.

• Identify the correct statements regarding subtracting integers.

  Hints

  Try comparing different combinations of adding and subtracting positive and negative numbers.

  Solution

  The statements are provided below:

  • 'Subtracting a positive number is the same as adding the negative of the number.' This definition is true. $4-(-4)$ is the same as $4+4$. Try it out!
  • 'Adding two numbers always results in a larger number.' This statement is false. If just one of the numbers is negative, this answer is not true. When we add $4+(-3)$, the answer, $1$, is smaller than $4$.
  • 'Subtracting two numbers always results in a larger number.' This statement is also false. While this is true if just one of the numbers is negative, this answer is not true. When we add $6-2$, the answer, $4$, is smaller than $6$.
  • 'Subtracting a negative number is the same as subtracting a positive number.' This answer is false. In the equation $8-(-1)$, subtracting a positive number will result in $7$, which is incorrect. When we add $8-(-1)$, $9$ is the correct answer.

• Identify the equivalent expressions.

  Hints

  Adding two positive integers always yields a positive sum while adding two negative integers always yields a negative sum.

  However, subtracting a negative number is like adding the same number.

  Solution

  Let's simplify the signs to find an equivalent equation.

  • If we have an equation $7 - (-1) + 3$, and we know that subtracting a negative number is the same as adding a positive number, we can simplify this equation to $7 + 1 + 3$.
  • For the equation $-7 + (-1) + (-3)$, we know that adding a negative number is the same as subtracting a positive number. So now we can simplify this equation to $-7 - 1 - 3$.
  • Given the equation $7 - 1 - (-3)$, if we remember that subtracting a negative number is the same as adding a positive number, we can simplify this equation to $-7 - 1 + 3$.
  • Lastly, for the equation $-7 - (-1) + 3$, we know that subtracting a negative number is the same as adding a positive number, so we can simplify this equation to $-7 + 1 + 3$.