Writing Systems of Inequalities
Real situations rarely demand an exact number — they set limits. A budget caps what you can spend, a fundraiser sets a minimum to earn, a truck has a maximum weight. Each limit translates into an inequality, and when a situation has two limits at once, the two inequalities together form a system.
Writing the system is a translation job: name your variables, find each limit in the words, and turn each one into its own inequality. The hardest part is choosing the right symbol, and a handful of key phrases decide that for you.
The phrase picks the symbol
"At most", "no more than", and "cannot exceed" all mean — the quantity can equal the limit or sit below it, but not pass it.
"At least", "no fewer than", and "a minimum of" all mean — the quantity can equal the limit or sit above it.
Notice that both families include the limit itself. If you can spend at most dollars, spending exactly dollars is allowed. That is why these phrases use and rather than the strict symbols and .
Each limit becomes its own inequality
Word problems for systems usually contain two separate limits: one about money (a cost or an earnings goal) and one about a count (how many items, hours, or trips). Handle them one at a time.
The money inequality multiplies each variable by its price or rate: if shirts cost dollars and hats cost dollars, the total spent on shirts and hats is . The count inequality usually adds the plain variables: the total number of items is .
The most common mistake is attaching the prices to the wrong variables or putting prices in the count inequality. Prices only belong in the money inequality; the count inequality has no coefficients other than .
Read each inequality back as a sentence
After writing the system, check it by translating each inequality back into English. Does say "the total cost is at most dollars"? Does say "there are at least items"? If your sentences match the problem, the system is right.
Worked examples
Example 1: a spending limit and an item minimum
A team buys shirts for dollars each and hats for dollars each. The team can spend at most dollars and needs at least items. Let be shirts and be hats. Write the system.
Answer: and
Example 2: an earnings goal and a time limit
Mia walks dogs for dollars each and washes cars for dollars each. She wants to earn at least dollars but has time for no more than jobs. Let be dogs walked and be cars washed. Write the system.
Answer: and
Example 3: one limit only
Jade can study at most hours this weekend — hours of math and hours of history. Write the inequality that models this limit.
Answer:
Try one yourself
Common questions
How do I decide between the two inequality directions?
Ask whether the number in the problem is a ceiling or a floor. Budgets, capacities, and time limits are ceilings, so the expression is the number. Goals, requirements, and minimums are floors, so the expression is the number.
Which numbers go in front of the variables?
Prices and rates. If small boxes weigh pounds and large boxes weigh pounds, the weight expression is . The counting inequality — how many boxes total — is just , with no coefficients.
Why is it a system and not one inequality?
Because the situation imposes both limits at the same time. A choice of and only works if it satisfies every condition, which is exactly what a system of inequalities means: all of its inequalities must be true at once.
Do I need to graph the system too?
Only if the problem asks. Writing the system and graphing it are separate skills — the graph's overlapping region shows all workable choices, and that step follows the same rules as any system of inequalities.
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