Problem 73 - College Algebra

Question

Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. \(-3.6\) and \(-1.4\)

Step-by-Step Solution

Answer

The distance between \(-3.6\) and \(-1.4\) is \(2.2\).

Step 1 : Set up the absolute value expression

The distance between \(-3.6\) and \(-1.4\) is given by \(|-3.6 - (-1.4)|\). The subtraction inside the absolute value becomes addition because we are subtracting a negative number, so we are actually adding 1.4 to -3.6.

Step 2: Simplify the expression inside the absolute value

Now we simplify the expression inside the absolute value: \(-3.6 + 1.4 = -2.2\). So now our absolute value expression looks like \(|-2.2|\).

Step 3: Evaluate the absolute value

Now we just need to find the absolute value of -2.2. The absolute value of a number is its distance from zero, so it is always positive or zero. So, \(|-2.2| = 2.2\).

Key Concepts

Distance Between Numbers

In mathematics, the distance between two numbers on the number line is often expressed using absolute value. This number represents how far apart the numbers are, ignoring their direction. Consider two numbers, like \(-3.6\) and \(-1.4\). When calculating the distance between them:

• First, write down both numbers. Determine which is larger and smaller if that helps simplify.
• Find the difference by subtracting one number from the other. The order doesn't matter because the distance is always positive.
• Use the subtraction of values as the expression inside the absolute value sign.

By expressing their difference using absolute value, \(|-3.6 - (-1.4)|\), we ensure the outcome reflects only the distance, stripping away any sign associated with direction on the number line.

Evaluating Expressions

Evaluating expressions involves performing mathematical operations that transform them into more straightforward, solvable formats. In this context, for the expression \(|-3.6 - (-1.4)|\), this process begins by tackling the operation within the absolute value.

• Recognize that subtracting a negative is equivalent to adding a positive. Hence, \(-3.6 - (-1.4)\) becomes \(-3.6 + 1.4\).
• This changes our original complex-looking expression into a simpler arithmetic operation.
• Ensure every step follows PEMDAS/BODMAS rules - Parentheses, Exponents (Orders), Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right).

Through carefully following these rules, you maintain accuracy in simplifying expressions, ultimately leading towards an understandable value.

Simplifying Expressions

Simplifying expressions is the act of rewriting expressions to make them easier to interpret and solve. With expressions involving absolute values, simplification usually means simplifying what's inside first.

• For the expression \(-3.6 + 1.4\), calculate the result of adding these two numbers, which yields \(-2.2\).
• This step ensures that you've reduced the problem to its simplest form inside the absolute value, \(|-2.2|\).
• After simplifying the inner operation, apply the absolute value to determine the positive magnitude.

By reducing expressions through simplifying, mathematical problems transition into clear and direct calculations, making further evaluations straightforward.

Negative Numbers

Negative numbers often lead students to question their effects in arithmetic operations. Understanding their properties becomes crucial in expressions involving subtraction and absolute values.

• Negative numbers represent values less than zero, marked by a minus sign.
• When subtracting a negative number, the result is essentially an addition. For example, \(-3.6 - (-1.4)\) translates to \(-3.6 + 1.4\).
• The absolute value of any number converts negatives to their positive distance from zero on the number line, such as \(-2.2\) to \(|-2.2| = 2.2\).

Gaining comfort with negatives means recognizing patterns in operations, ultimately aiding in simplifying and evaluating mathematical expressions efficiently.