Problem 88 - Beginning and Intermediate Algebra

Question

Simplify. $$ \frac{-15}{1-4} $$

Step-by-Step Solution

Answer

The simplified form is 5.

Step 1: Determine the Denominator

Start by simplifying the denominator. The expression given is \(1 - 4\). Calculate this by subtracting 4 from 1, which equals \(-3\).

Step 2: Simplify the Fraction

Now that we have found the denominator to be \(-3\), we need to simplify the overall fraction. The fraction is \(-15\) divided by \(-3\).

Step 3: Perform Division

Perform the division by calculating \(-15 \div -3\). When two negative numbers are divided, the result is a positive number. Thus, \(-15 \div -3 = 5\).

Key Concepts

Negative Numbers

Understanding negative numbers is essential when working with algebraic expressions and fractions. Negative numbers are those found less than zero on the number line. They often appear with a minus (-) sign.

• For example, -1, -15, and -3 are all negative numbers.
• In mathematical operations, negative numbers have unique rules that differentiate them from positive numbers.

When adding or subtracting negative numbers, it's like moving left on the number line. For multiplication and division, two negatives make a positive. This is crucial to understand when simplifying fractions like \(-15 \div -3\). The result of dividing or multiplying two negative numbers is always positive.

Division

In mathematics, division is one of the four basic operations, besides addition, subtraction, and multiplication. It is the process of finding out how many times one number is contained within another. When you divide a number by another, you're splitting it into equal parts. The number you're dividing is called the "dividend", and the number you divide by is the "divisor".

• For example, in the expression \(-15 \div -3\), -15 is the dividend and -3 is the divisor.
• When dividing two negative numbers, the division result is positive. Thus, \(-15 \div -3 = 5\).

This knowledge is important when you simplify fractions involving negative numbers and when you handle more complex algebraic expressions.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables (letters that represent numbers), and operators (such as +, -, \( imes\), and \(\div\)). These expressions can be simplified or evaluated by performing operations according to mathematical rules.

• For instance, \(1 - 4\) is a simple algebraic expression where variables are not present, and it combines numbers with subtraction.
• In the expression \(-\frac{15}{1-4}\), simplifying \(1 - 4\) results in \(-3\), forming a fraction \(-\frac{15}{-3}\), which simplifies further by applying division.

Mastering algebraic expressions involves simplifying them by reducing similar terms or applying arithmetic operations. This can turn complex expressions into more manageable forms, easing the way for problem-solving and analysis.