Problem 74 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ -5[(-1+5)+(6-8)] $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression -5[(-1 + 5) + (6 - 8)]. Answer: -10

Step 1: Solve inside the brackets

First, we need to solve the expression inside the brackets. We have two operations: (-1 + 5) and (6 - 8). Let's calculate them: $$ (-1 + 5) = 4 \\ (6 - 8) = -2 $$

Step 2: Substitute results back into the expression

Now, let's replace (-1 + 5) by 4 and (6 - 8) by -2 in the original expression: $$ -5[(4) + (-2)] $$ Now, we still have one operation inside the brackets, which is the addition of 4 and -2. So, let's calculate this operation: $$ (4 + (-2)) = 2 $$

Step 3: Multiply the result by -5

We now have the simplified expression without brackets: $$ -5 \times 2 $$ To find the final value, we simply multiply -5 by 2: $$ -5 \times 2 = -10 $$ So, the value of the given expression is -10.

Key Concepts

Brackets in Algebra

Brackets play a crucial role in algebra. They help us group parts of an expression that need to be solved first. When faced with a complex expression, always start with the innermost brackets. This ensures that operations are performed in the correct order, following the order of operations rules.
In our exercise, the expression is \(-5[(-1+5)+(6-8)]\). Look inside the brackets first. You need to address each operation within them before moving outward:

• Calculate inside each pair of brackets: \((-1+5)\) and \((6-8)\).
• Once these results are found, substitute them back into the expression.

This step-by-step approach allows you to simplify the problem and handle each operation correctly. This method of simplifying expressions using brackets is essential in all areas of mathematics.

Negative Numbers

Handling negative numbers can be tricky, but understanding a few simple rules can make it easier. In math, negative numbers are used to represent a value less than zero. They are especially important when performing subtraction or dealing with opposite values.
In the example, we dealt with both addition and subtraction involving negative numbers:

• \((-1 + 5) = 4\): Adding a positive number to a negative number means you move towards the positive.
• \((6 - 8) = -2\): Subtracting a larger number from a smaller one results in a negative number.

Knowing how negative numbers interact with each other can help avoid mistakes. When multiplying, like in \(-5 \times 2\), remember that a negative times a positive will always give a negative result. Practicing these rules will boost your confidence in dealing with negative numbers in algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can look complex but break them down step by step to simplify.
In this example, the expression is \(-5[(-1+5)+(6-8)]\). Treat every part individually:

• The operations within the inner brackets are evaluated first.
• Substitute back to see a simpler expression, like \(-5[(4)+(-2)]\).
• Solve the remaining operations, such as addition, inside brackets.

By focusing on each piece of the expression one at a time, you can simplify it efficiently. Understanding how to manipulate algebraic expressions is a core skill in math that allows you to solve complex equations and understand the relationships between different numbers and variables.