Linear Word Problems and Inequalities
Translate the words, watch the flip, answer what they asked
Words into math, one phrase at a time
Word problems are translation exercises. Build the dictionary: 'of' means multiply, 'per' marks a rate that becomes your slope, 'is' means equals, 'at least' means greater than or equal, 'at most' and 'no more than' mean less than or equal, and 'less than' flips the order, so '7 less than x' is x - 7, not 7 - x.
Define variables with units and a time frame before writing any equation. Let m = the number of months beats let m = money every time. Half of all word-problem errors are really variable-definition errors.
The two-equation setup
Ticket, coin, and mixture problems all use the same skeleton: a count equation and a value equation. If a adult and c child tickets sell, then a + c = total tickets, and (adult price)a + (child price)c = total revenue. Two equations, two unknowns, one intersection.
Any two-unknown word problem becomes a Desmos problem the moment you write the equations. Rename the variables x and y, type both lines, and tap the intersection. You can also test an inequality by typing it into Desmos and plugging candidate values, or by graphing it and looking at the shaded region.
Inequalities: one rule with teeth
Inequalities behave exactly like equations with a single exception: multiplying or dividing both sides by a negative number flips the inequality sign. Adding and subtracting never flip anything. When you solve -3x > 15, the answer is x < -5, and the flipped sign is the entire point of the question.
Two classic traps. First, answer the quantity that was asked: if you solved for adult tickets and they asked for child tickets, the wrong answer is sitting in the choices waiting for you. Second, respect rounding direction: a budget that allows 12.5 months of membership buys only 12 whole months.
Rounding direction in one line: capacity questions round down (how many boxes fit) and requirement questions round up (how many trips are needed). The context, not the arithmetic, decides.
When translation feels tangled, backsolve. Take a middle answer choice, push it through the story, and see whether the total comes out high or low. Two plugs usually corner the answer, and the story itself checks your work.