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Translate word problems into inequalities and solve with step-by-step solutions
Word problems require translating English phrases into mathematical symbols. Here are the most common translations for inequality word problems.
Greater Than or Equal (≥)
Less Than or Equal (≤)
Greater Than (>)
Less Than (<)
"At least" means ≥ (greater than or equal), including the value. "More than" means > (strictly greater), not including the value. "At least 5" includes 5, "more than 5" doesn't.
Focus on the key phrase: "at least" and "minimum" point up (≥), while "at most" and "maximum" point down (≤). "More than" points up (>), "less than" points down (<).
"No more than" means ≤ (less than or equal to). It's the same as "at most". For example, "no more than 10" means x ≤ 10.
Yes! Phrases like "between 5 and 10" create compound inequalities: 5 < x < 10. "At least 5 but no more than 10" becomes 5 ≤ x ≤ 10.
Substitute your answer back into the context of the problem. If the problem says "at least 50 points to pass" and you got x ≥ 45, that doesn't make sense. Always verify with the original context.
When dividing by a negative coefficient to solve, remember to flip the inequality sign. This is true even in word problems!
"Exceeds" means "is greater than" (>). For example, "sales exceed $1000" translates to sales > 1000, not including exactly $1000.
"Within" typically creates an absolute value inequality. "Within 5 of 20" means |x - 20| ≤ 5, which translates to 15 ≤ x ≤ 25.