When comparing two numbers, we have two fundamental methods. The subtraction method compares the difference a minus b with zero. If a minus b is positive, then a is greater than b. The division method compares the ratio a over b with one, provided b is positive. These visual representations on the number line help us understand these comparison methods clearly.
The subtraction method is systematic. First, we calculate the difference between the two expressions. Then we simplify and analyze when this difference is positive, negative, or zero. In our example, we compare 3x squared plus 2 with 2x squared plus 5. The difference is x squared minus 3. This tells us that the first expression is larger when x squared is greater than 3, and smaller when x squared is less than 3.
The division method is particularly useful when comparing positive quantities. We divide one expression by another and compare the result with 1. In our example, we compare 2 to the power x with 3 to the power x. Their ratio is 2 thirds to the power x. Since 2 thirds is less than 1, this ratio decreases as x increases, confirming that 2 to the power x is always less than 3 to the power x when x is positive.