Problem 47 - Beginning and Intermediate Algebra

Question

Simplify each expression. (Remember the order of operations.) See Examples 4 and 5. $$ -6-(2-11) $$

Step-by-Step Solution

Answer

The simplified expression is 3.

Step 1: Deal with Parentheses

The expression inside the parentheses should be addressed first. Here, you have the expression \(2 - 11\). Subtract \(11\) from \(2\) to get \(-9\).

Step 2: Substitute the Parentheses Result Back

Once you calculate \(2 - 11\) as \(-9\), substitute \(-9\) back into the original expression. The expression becomes \(-6 - (-9)\).

Step 3: Handle the Double Negative

When you have a negative sign outside a parenthesis with a negative number inside, they cancel each other out to become positive. Therefore, \(-6 - (-9)\) becomes \(-6 + 9\).

Step 4: Perform the Addition

Now, add \(-6\) and \(9\) together. This is just like subtracting \(6\) from \(9\). The result is \(3\).

Key Concepts

Simplifying Expressions

Simplifying expressions might look tricky, but it is the foundation of algebra. When you simplify an expression, the idea is to make it smaller and easier to understand or calculate while maintaining its original value. It’s like cleaning up a room – you have to know what to keep and what can go.

• The first step in simplifying is to follow the order of operations, which ensures that you solve problems consistently.
• You tackle operations inside parentheses first, then handle exponents, followed by multiplication and division from left to right, and finally, addition and subtraction from left to right. This is often remembered with the acronym PEMDAS.

In our example, it all starts with simplifying what's inside the parentheses. Once that's done, you look outside to further simplify the expression by combining like terms or handling operations that remain. This step-by-step method ensures that the result is accurate every time.

Handling Negative Numbers

Negative numbers can be a bit intimidating at first, but with some practice, handling them becomes second nature. They follow the same rules as positive numbers, with a twist.

• When you subtract a larger number from a smaller one, the result goes below zero and becomes negative, like in the case of 2 - 11 = -9.
• Additionally, when dealing with a double negative – such as subtracting a negative number – the negatives cancel each other out and the result turns positive.

For instance, if you see an expression like -6 - (-9), you change it to -6 + 9 because two negative signs together make a positive. Understanding these basics allows you to navigate through more complex math problems with confidence.

Parentheses in Math

Parentheses play a crucial role in mathematics, helping clarify which operations to perform first. Solving equations accurately often depends on dealing with parentheses correctly.

• When you see parentheses, they signal that the operations inside should be completed first before moving on to other parts of the expression, as dictated by the order of operations.
• This is why solving 2 - 11 inside the parentheses was the priority in our exercise example.

The result from the parentheses is then replaced into the original expression, allowing further simplification. Parentheses ensure expressions are interpreted as intended and help avoid mistakes that arise from performing operations out of order.