Problem 70 - Elementary Algebra

Question

Simplify. $$ 3-7|-2-3|+43 $$

Step-by-Step Solution

Answer

The expression simplifies to 11.

Step 1: Solve the Absolute Value

First, identify the expression inside the absolute value: \(-2-3\). Simplify this by combining the numbers: \(-2-3 = -5\). Then, calculate the absolute value:\(|-5| = 5\).

Step 2: Substitute and Rearrange

Substitute the calculated absolute value back into the original expression:\(3 - 7 imes 5 + 43\). Use multiplication to simplify:\(3 - 35 + 43\).

Step 3: Simplify the Expression

Begin simplifying by performing the operations from left to right: First, add 3 and 43:\(3 + 43 = 46\). Then subtract 35:\(46 - 35 = 11\).

Key Concepts

Understanding Absolute Value

Absolute value is a way to describe how far a number is from zero on the number line, regardless of direction. For any real number, whether positive or negative, the absolute value is always non-negative.

For instance, the absolute value of \(-5\) is \(|-5| = 5\). This is because \(-5\) is five units away from zero on the number line. Similarly, for \(+5\), the absolute value is also \(|5| = 5\). The concept simplifies expressions by transforming any number inside the absolute value signs to its positive distance from zero.

In algebraic expressions, it is crucial to perform any calculations inside the absolute value bars before resolving absolute value itself. This ensures accurate simplification and helps in evaluating expressions correctly.

Steps in Substitution and Rearrangement

Substitution involves replacing a value or expression with a given equivalent. Once you've calculated part of the expression, like an absolute value, substitution allows you to insert that calculated value back into the expression to move towards simplification.

In our example, after solving the absolute value \(|-5| = 5\), the next step is substituting \(-2-3\) with the value \(+5\) into the expression, rewriting it as \((3 - 7 \times 5 + 43)\). This substitution is crucial because it simplifies the expression step by step.

Rearranging is often necessary when dealing with expressions. After substitution, always check if the expression can be rearranged to make calculations straightforward. This process helps in maintaining clarity and ensuring you don’t miss any operations.

Mastering the Order of Operations

The order of operations is a set of rules that dictate the sequence in which computations are performed. The conventional order, remembered as PEMDAS, stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Applying this rule ensures consistent results. In our problem, after substituting the absolute value, the expression becomes \((3 - 7 \times 5 + 43)\).

• First, solve multiplication: \(7 \times 5 = 35\).
• Then, proceed to addition and subtraction from left to right: \(3 - 35 + 43\).
• Combine the remaining terms to get: \(3 + 43 = 46\), then \(46 - 35 = 11\).

Understanding and following the order of operations ensures that expressions are simplified accurately and efficiently.