Problem 114 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. 19 less than the difference of 9 and -2

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Understand the problem

Identify the mathematical operations described in the phrase. 'Quotient' indicates division, and 'sum' indicates addition.

Step 2: Translate the phrase into a numerical expression

First, translate the sum part of the phrase. The sum of -8 and -2: \[-8 + (-2)\]

Step 3: Simplify the sum

Add -8 and -2 together: \[-8 + (-2) = -10\]

Step 4: Form the quotient expression

Form the quotient of -20 and the simplified sum: \[-20 \div (-10)\]

Step 5: Simplify the quotient

Divide -20 by -10: \[-20 \div (-10) = 2\]

Key Concepts

Division

Division is one of the most basic operations in mathematics. It can be thought of as the process of sharing a number into equal parts. In a division statement, the number you want to divide is called the 'dividend,' and the number you are dividing by is called the 'divisor.' The result is called the 'quotient.'
For example, in this exercise, -20 is the dividend and -10 is the divisor. When you divide -20 by -10, you get the quotient 2.
Always remember the rule for dividing signs:

• If both numbers have the same sign, the quotient is positive.
• If the numbers have different signs, the quotient is negative.

In our current exercise, both the dividend and the divisor are negative, hence the quotient is positive.

Addition

Addition is another fundamental operation where you combine two or more numbers to get a sum. When you add negative numbers, you add their absolute values and then place a negative sign before the result.
In the given exercise, you need to add -8 and -2. So, you add their absolute values: 8 and 2. This gives you 10. Since both original numbers are negative, the sum is -10.
In general:

• When adding two positive numbers, the result is positive.
• When adding two negative numbers, the result is negative.
• When adding a positive and a negative number, the result takes the sign of the number with the greater absolute value.

This understanding is essential when simplifying numerical expressions.

Simplification

Simplification involves performing the indicated operations to reduce a numerical expression to its simplest form. This process makes the problem easier to work with and understand.
In our exercise, the phrase was given in words, and we translated it into operations: division and addition.
Steps for simplification:

• Identify and translate all operations indicated by the phrase. For example, 'the sum of -8 and -2' translates to \[-8 + (-2)\].
• Perform the addition operation first: \[-8 + (-2) = -10\].
• Use the result from step 2 in the division operation: \[-20 \div (-10)\].
• Finally, perform the division to find the simplified form: \[-20 \div (-10) = 2\].

By following these steps, you can tackle any similar problems easily and systematically.