Problem 45 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{-56+h}{-8} $$

Step-by-Step Solution

Answer

The simplified expression is \(7+\frac{h}{8}\).

Step 1: Analyze The Expression

Observe the given expression \(-\frac{56+h}{8}\), where 8 is the negative divisor.

Step 2: Simplifying the Expression part 1

Divide each term separately by -8. The fraction simplifies as \(-\frac{56}{-8} - \frac{h}{-8}\) , which is equivalent to 7-\(-\frac{h}{8}\).

Step 3: Simplifying the Expression part 2

Now simplify the second term. The negative sign in the numerator and denominator cancels out. Therefore, h divided by -8 simplifies as \(-\frac{h}{8} = \frac{h}{8}\).

Step 4: Final Simplified Expression

Substitute back the simplified term into the previous expression. The final simplified expression is \(7+\frac{h}{8}\).

Key Concepts

Negative Numbers

Understanding how to work with negative numbers is essential in algebra. A negative number is any number that is less than zero. In mathematics, negative numbers are indicated with a minus sign (-).
When working with negative numbers, remember that:

• Multiplying or dividing two negative numbers results in a positive number. For example, \[-8 \times -7 = 56\]
• Multiplying or dividing a positive number with a negative number results in a negative number, e.g., \[4 \times -2 = -8\]
• Adding a negative number is the same as subtracting it. For instance, \[5 + (-3) = 5 - 3 = 2\]

In our exercise, we used a negative divisor in the fraction. This means each part of the fraction—both the number and any variables—is divided by a negative value. This can change signs as shown in the process of simplification.

Fraction Division

Fraction division might seem complicated, but breaking it down step by step can simplify the process. Dividing fractions involves dividing each term of a larger expression separately if possible.
Key points to remember:

• When dividing by a fraction, multiply by its reciprocal. For example, dividing by \[\frac{1}{2} \] is equivalent to multiplying by \[2\]
• When dividing terms within an algebraic fraction independently, divide each term by the denominator separately.
• If both the numerator and denominator are negative (as in our exercise), the negative signs will cancel each other out, resulting in a positive term.

In our example, we turned the expression into two separate terms and divided each by the negative divisor, which helped to clear the negative signs from the equation.

Expression Simplification

Simplifying expressions is a fundamental algebra skill that makes handling complex equations manageable. It involves reducing an expression to its simplest form.
Steps to simplify expressions:

• Break down the terms: In the exercise, we split the fraction into separate components \[-\frac{56}{-8} \quad \text{and} \quad \frac{h}{-8}\]
• Simplify each term: This means performing basic math operations like division. \[ -\frac{56}{-8} = 7\]
• Combine similar terms: Here, we consolidated the simplified terms to form a more straightforward expression.
• Be mindful of negative signs: Cancel wherever applicable, as done in \[-\frac{h}{8} \] becoming \[\frac{h}{8}\]

By following these steps, the final expression from the exercise becomes much simpler: \[7 + \frac{h}{8}\] which completes our simplification process.