Problem 75 - Elementary Algebra

Question

Simplify. $$ (-4) 3-(2-|-4|) \div||-32+||7 $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{79}{16}\).

Step 1: Evaluate Absolute Values

Begin by evaluating the absolute values. We have three occurrences: - \( |-4| = 4 \)- \( |-32| = 32 \)- \( ||-32+|| = 32 \) since the inner absolute is considered on its own before the operation.The expression now looks like \((-4) \cdot 3 - (2-4) \div 32 + 7\).

Step 2: Simplify Inside Parentheses

Simplify the expression inside the parentheses: \( 2 - 4 = -2 \)The expression is now \((-4) \cdot 3 - (-2) \div 32 + 7\).

Step 3: Perform Multiplication

Multiply \(-4\) by \(3\): \((-4) \cdot 3 = -12 \)Substitute this into the expression: \(-12 - (-2) \div 32 + 7\).

Step 4: Perform Division

Divide \(-2\) by \(32\): \((-2) \div 32 = -\frac{1}{16}\)Consider this result for the next calculation: \(-12 - (-\frac{1}{16}) + 7\).

Step 5: Simplify Subtraction and Addition

Subtracting a negative is the same as adding its positive, so: \(-12 + \frac{1}{16} + 7\).Combine like terms: \((-12 + 7 = -5)\),Now simplify: \(-5 + \frac{1}{16}\).

Step 6: Final Addition

Convert \(-5\) to a fraction with a denominator of \(16\) to combine with \(\frac{1}{16}\): \(-5 = -\frac{80}{16}\),Now perform the addition: \(-\frac{80}{16} + \frac{1}{16} = -\frac{79}{16}\).

Key Concepts

Absolute Value

The absolute value of a number essentially measures its distance from zero on a number line. It is always a non-negative number. In our expression, you see several instances of absolute values.
For example:

• When you see \(|-4|\), it becomes \(4\) because \ |*-4*| = 4\.
• The absolute value of \(-32\) is \(32\). This is because \(|-32|\) equals the positive 32.
  Always turn negative numbers inside the bars into positives.
• Double absolute value signs like \(||-32||\) are treated the same way. Just focus on the innermost absolute value.

Our expression uses absolute value to simplify terms. This process can make calculations easier by reducing operations in later steps.

Order of Operations

The order of operations is crucial for solving algebraic expressions correctly. It tells you the sequence to follow:
Remember the acronym PEMDAS:

• **P**: Parentheses first. Look inside them and simplify.
• **E**: Exponents follow next, though not in this example.
• **MD**: Multiplication and Division come in order from left to right. No priority among them—as you encounter them, do them.
• **AS**: Addition and Subtraction perform last, again from left to right.

In our situation, we processed the absolute values first, and then managed any expressions inside parentheses.
Always follow the sequence to avoid mistakes and confusion.

Fractions

Fractions represent parts of a whole and can often appear in algebraic expressions. They consist of a numerator on top and a denominator on the bottom. Considerations may include:

• Always try to simplify fractions where possible to make operations easier.
• In our example, \(-2\div32\) simplifies as \(\frac{-1}{16}\). Do this division before proceeding further.
• When adding or subtracting fractions, ensure they have common denominators by converting them if necessary.
• Notice in the final step, \(-5\) became \(\frac{-80}{16}\), so it could add to \(\frac{1}{16}\).

Understanding fractions and their manipulations is key to dealing with rational numbers in expressions.

Simplifying Expressions

Simplifying expressions means reducing it to its simplest form. Here is what you need to consider:

• Use operations like addition, subtraction, multiplication, and division to combine terms.
• Simplifying involves following the order of operations and managing all steps methodically.
• In this problem, after conducting multiplication and division, ensure all terms are in their simplest form. This includes combining like terms.
• Be sure to convert \(-5\) to a common fraction when dealing with fractions to finalize the expression. This step is crucial to ensuring accuracy in the result, as seen when converting to \(\frac{-79}{16}\).

By following these steps and principles, you'll be able to simplify any expression effectively.