What is the solution to Three-fourths a greater-than negative 16? a greater-than negative 21 and one-third a less-than negative 21 and one-third a greater-than 21 and one-third a less-than 21 and one-third
The Correct Answer and Explanation is:
Let’s break down and solve the inequality:
Given:
Three-fourths a > -16
This is written as:
(3/4)·a > -16
Step 1: Solve the inequality
To isolate a, multiply both sides of the inequality by the reciprocal of 3/4, which is 4/3: a>−16⋅43a > \frac{-16 \cdot 4}{3} a>−643a > \frac{-64}{3}
So the solution is:
a > -64/3
Step 2: Interpret the answer
Now convert -64/3 into a mixed number: −64÷3=−21 remainder 1⇒−2113-64 \div 3 = -21 \text{ remainder } 1 \Rightarrow -21 \frac{1}{3}
So:
a > -21⅓
This means the value of a must be greater than -21⅓.
Step 3: Choose the correct option
Now evaluate the given choices:
- a > -21⅓ ✅
- a < -21⅓ ❌
- a > 21⅓ ❌
- a < 21⅓ ❌
Only one choice correctly represents the solution.
Final Answer:
a > -21 and one-third
Explanation
To solve the inequality “Three-fourths a greater than -16”, we begin by translating this into algebraic form: (3/4)·a > -16. The goal is to isolate the variable a so we can determine which values satisfy the inequality. Since a is being multiplied by 3/4, we eliminate this fraction by multiplying both sides by the reciprocal of 3/4, which is 4/3. This operation is valid because multiplying both sides of an inequality by a positive number does not change the direction of the inequality.
So, we multiply both sides by 4/3:
(4/3) × (3/4)·a > (4/3) × -16.
This simplifies to: a > -64/3.
The fraction -64/3 is then converted into a mixed number for easier understanding. Dividing 64 by 3 gives 21 with a remainder of 1, so -64/3 is equal to -21 and 1/3 (or -21⅓). Therefore, the final solution is a > -21⅓.
We then examine the multiple-choice options to find which one correctly matches our solution. The only option that expresses a greater than -21⅓ is a > -21 and one-third. All other options either incorrectly reverse the inequality sign or point to positive values of 21⅓, which are irrelevant to the actual solution.
Thus, the correct answer is a > -21 and one-third.
