Problem 11 - Intermediate Algebra

Question

If each of the following expressions were evaluated, what would be the sign of the result? a. \(-1,763+1,699\) b. \(-503-512\) c. \((-657)(-22)\) d. \(\frac{-2,744}{49}\)

Step-by-Step Solution

Answer

a. Negative, b. Negative, c. Positive, d. Negative.

Step 1: Evaluate Expression a

For the expression \(-1,763 + 1,699\), calculate the sum to find the sign. The result of the addition is \(-1,763 + 1,699 = -64\). The sign of \(-64\) is negative.

Step 2: Evaluate Expression b

For the expression \(-503 - 512\), calculate the sum by adding the absolute values and keeping the negative sign. The result is \(-503 - 512 = -1,015\). The sign of \(-1,015\) is negative.

Step 3: Evaluate Expression c

For the expression \((-657)(-22)\), multiply the two negative numbers. The product of two negative numbers is positive. Thus, \((-657)(-22)\) results in a positive sign.

Step 4: Evaluate Expression d

For the expression \(\frac{-2,744}{49}\), divide a negative number by a positive number. The result of the division is a negative number as the number \(-2,744\) divided by \(49\) will maintain the negative sign.

Key Concepts

Addition of Signed Numbers

When adding signed numbers, it's important to pay attention to the signs of the numbers involved. You can follow these general rules for a better understanding:

• If both numbers have the same sign, add their absolute values. The result will keep the common sign. For instance, \(-503 - 512\) adds the magnitudes \(503 + 512 = 1015\), keeping the negative sign, resulting in \(-1015\).
• If the numbers have different signs, subtract the smaller absolute value from the larger one. The result will have the sign of the number with the larger absolute value. An example would be evaluating \(-1,763 + 1,699\) where you calculate \(1,763 - 1,699 = 64\). Since \(1,763\) is negative, the outcome is \(-64\).

Once you grasp these basics, adding signed numbers becomes predictable and much easier to manage.

Multiplication of Signed Numbers

Multiplying signed numbers can seem tricky, but it follows straightforward rules that once mastered can simplify the process:

• When both numbers are positive, the result is positive. For example, \(3 \times 2 = 6\).
• When both numbers are negative, the result is also positive, since two negatives negate each other. For instance, calculating \((-657)(-22)\) results in a positive sign because two negative factors result in a positive product. Thus, the product is positive.
• If one number is negative and the other is positive, the result is negative. This is because only one factor remains negative, affecting the overall product. For example, \((-7) \times 5 = -35\).

Mastering these rules can greatly aid in managing and solving multiplication problems involving signed numbers.

Division of Signed Numbers

The process of division for signed numbers is conceptually similar to multiplication. The sign of the result depends on the signs of the dividend and the divisor:

• If both the dividend and divisor are positive, the quotient is positive. For example, \(18 \div 6 = 3\).
• If both the dividend and divisor are negative, the quotient is still positive, as the negatives cancel each other out. For instance, \((-20) \div (-5) = 4\).
• If one is negative and the other is positive, the quotient is negative. For example, in \(\frac{-2,744}{49}\), the result is negative because a negative dividend divided by a positive divisor yields a negative quotient.

Always remember these rules to quickly determine the signs of results when dividing signed numbers. It makes complex calculations more straightforward and manageable.