Problem 79 - Intermediate Algebra

Question

Perform the operations and simplify. $$ 10(h-9) \frac{h-3}{9-h} $$

Step-by-Step Solution

Answer

The simplified expression is \( -10h + 30 \).

Step 1: Distribute the 10

Begin by expanding the expression by distributing 10 into the expression inside the parentheses: \[ 10(h-9) = 10h - 90 \].The expression becomes:\[ (10h - 90) \frac{h-3}{9-h} \].

Step 2: Recognize the Negative

Observe that the denominator \[ 9-h \] can be rewritten for simplification. Notice that \[ 9 - h = -(h - 9) \].This expression can be recast as \[ -(h-9) \].

Step 3: Simplify the Fraction

Substituting the observation from Step 2 into the expression, the fraction becomes:\[ \frac{h-3}{-(h-9)} = -\frac{h-3}{h-9} \].

Step 4: Apply Simplified Denominator

Now, substitute this into the main expression:\[ (10h - 90) \left(-\frac{h-3}{h-9}\right) \].Multiply across:\[ -(10h - 90) \frac{h-3}{h-9} = (90 - 10h) \frac{h-3}{h-9} \].

Step 5: Factor and Simplify Further

Factor the expression \( 90 - 10h \) as follows:\[ -10(h-9). \]This makes the expression:\[ -10(h-9) \frac{h-3}{h-9} \].

Step 6: Cancel Common Terms

Cancel out the common factor \( h-9 \) from the numerator and denominator in the expression:\[ -10 \frac{h-3}{1} = -10(h-3) \].Then distribute the \( -10 \):\[ -10h + 30 \].

Key Concepts

Understanding the Distributive Property

The distributive property is a vital tool in algebra. It lets you multiply a single term by each term within a parenthesis. Consider the scenario: you have the expression

• 10(h-9).

Here, we distribute "10" across each term inside the parentheses, which transforms into

• 10h - 90.

By doing this, we've successfully rewritten an expression that’s easy to work with for further algebraic operations.
When you use the distributive property, remember to keep track of positive and negative signs. Thus, if the number outside the parenthesis is negative, like

• -5(x - 2),

you multiply it to get

• -5x + 10.

Simplifying Fractions Effortlessly

Simplifying fractions is a technique that helps make expressions more manageable. When we have a fraction, like

• \( \frac{h-3}{9-h} \),

we should first look for opportunities to rewrite or simplify the terms. Notice how the denominator, \( 9-h \), can be rewritten as

• \( -(h-9) \).

This allows us to simplify the fraction as

• \( -\frac{h-3}{h-9} \).

After this simplification, the fraction fits neatly into the larger expression. Simplifying fractions is all about looking for easy transformations that can make your work easier.
You always want to aim for the simplest form possible to make further operations less complex and error-prone.

Mastering Factoring Expressions

Factoring expressions is crucial in algebra because it allows simplification and resolving complex equations. Consider the expression we worked with:

• 90 - 10h.

To factor this, notice that 10 is a common factor, which produces

• -10(h-9).

Factoring like this is about finding common factors that can be 'pulled out' to simplify the expression.
Once factored, you often reveal common terms in the numerator/denominator, allowing cancellation, simplifying further. For instance, having factored to

• -10(h-9)\frac{h-3}{h-9},

cancelling \( h-9 \) leads to a much simpler

• -10(h-3).

Factoring efficiently reduces the clutter and focuses on the essential parts of an expression.