A student club must raise more than $600 but no more than $1,500 to cover the cost of a field trip. Which inequality represents this situation, where is a possible amount, in dollars, the club could raise?
For inequality translation questions, move slowly through the wording and convert each phrase directly into symbols: “more than” means a strict inequality (), “less than” also is strict (), while “at least” and “no more than” include the endpoint ( or ). Identify the lower and upper limits separately, decide whether each endpoint is included or excluded, then write a compound inequality that satisfies both conditions at once and match it to the answer choices. This careful word-to-symbol matching is much faster and more reliable than trying to reason by plugging in many example numbers.
Hints
Focus on the phrase about
Look only at the words around : does “more than ” allow to be equal to , or must it be greater than ?
Focus on the phrase about
Now look at the words around : when something is “no more than ,” is itself allowed, or not allowed?
Combine both limits
You need an inequality that keeps between and while matching which endpoints are allowed. Which choice shows being bigger than but not bigger than ?
Desmos Guide
Graph the target range in your mind
Picture the description as numbers on a number line: amounts greater than but no more than . This is the interval just to the right of up through and including .
Graph each choice in Desmos
In Desmos, use instead of . For each answer choice, type it as an inequality in , for example:
x >= 1500600 <= x <= 1500600 < x <= 1500x < 600andx > 1500(enter these as two separate lines) Look at the shaded regions on the -axis that each choice produces.
Match the shading to the situation
Compare the shading for each choice to the mental picture from Step 1: values between and , not including but including . The choice whose shaded region exactly fits that description is the correct inequality.
Step-by-step Explanation
Translate “more than ”
Focus on the lower amount: the club must raise more than dollars.
- “More than ” means the amount must be greater than .
- This is written as (not including itself).
Translate “no more than ”
Now look at the upper amount: the club must raise no more than dollars.
- “No more than ” means can be or any smaller amount (as long as it still satisfies the first condition).
- This is written as (including itself).
Combine the two conditions
Both conditions must be true at the same time:
We write this as a compound inequality:
Now check the choices and select the one that exactly matches this compound inequality.