ombining them, rather than trying to do it all at once.","body":["Sam has $45. He spends 2/5 of it on a book (a fraction OF the current amount, so multiply first: 45 x 2/5 = 18, then subtract: 45 - 18 = 27), then earns $12 doing chores (a one-time change, add it on: 27 + 12 = 39)."],"worked":{"prompt":"Sam has $45. He spends 2/5 of it on a book, then earns $12 doing chores. How much money does he have now?","steps":["Amount spent on the book: 45 x 2/5 = 18.","Remaining after the book: 45 - 18 = 27.","After earning $12 from chores: 27 + 12 = 39."],"answer":"Sam has $39 now."},"checks":["Why is '2/5 of it' calculated using the ORIGINAL $45, not some other amount?","What operation models spending 2/5 of an amount: addition, subtraction, or multiplication?"]},{"id":"equations-inequalities","title":"4. Solving equations and inequalities to model real problems","phase":"Abstract","intro":"Writing then solving an equation or inequality follows the same undo-in-reverse-order method from earlier grades, now with rational coefficients allowed, plus one new rule for inequalities: dividing or multiplying both sides by a NEGATIVE number flips the inequality symbol.","body":["Equation: 'Three less than twice a number is 19' becomes 2n - 3 = 19. Undo in reverse order: add 3 to both sides (2n = 22), then divide by 2 (n = 11).","Inequality: a parking garage charges $5 plus $2 per hour; Jax has at most $17. Modeled as 5 + 2h <= 17, solving gives 2h <= 12, so h <= 6: he can park for at most 6 hours.","When the coefficient being divided out is negative, the inequality symbol flips. Solving -4x > 20 by dividing both sides by -4 gives x < -5, not x > -5."],"figures":[{"kind":"numberLine","min":-8,"max":2,"step":1,"marks":[{"value":-5,"label":"x = -5 (open circle)"}],"caption":"The solution to -4x > 20 is x < -5: every value strictly to the left of -5 makes the inequality true."}],"worked":{"prompt":"Solve -4x > 20 for x.","steps":["Divide both sides by -4. Because -4 is negative, the inequality symbol flips from > to <.","20 / -4 = -5, so x < -5.","Check with a value in the solution set, x = -6: -4 x (-6) = 24, and 24 > 20 is true."],"answer":"x < -5."},"checks":["Why does dividing both sides of an inequality by a negative number flip the symbol?","For 5 + 2h <= 17, why is the first step to subtract 5 rather than divide by 2?"]}],"guidedPractice":[{"question":"Simplify: 2/3 x + 4 - 1/3 x - 1","answer":"1/3 x + 3, because 2/3 x - 1/3 x = 1/3 x, and 4 - 1 = 3."},{"question":"Expand: 5(2x - 3)","answer":"10x - 15."},{"question":"Factor: 12x + 18","answer":"6(2x + 3), because the greatest common factor of 12 and 18 is 6."},{"question":"Write two equivalent expressions for a price p decreased by 15%, and simplify one into a single term.","answer":"p - 0.15p, which simplifies to 0.85p (the sale price is always 85% of the original)."},{"question":"Maria has $60. She spends 1/3 of it on a game, then her aunt gives her $15. How much does she have now?","answer":"$$55, because she spends 60 x 1/3 = $20 (leaving $40), then 40 + 15 = $55."},{"question":"Solve: 4n + 7 = -9","answer":"n = -4, because 4n = -16, so n = -16 / 4 = -4."},{"question":"Solve the inequality: -4x > 20","answer":"x < -5, because dividing both sides by -4 (negative) flips the symbol."}],"assessmentQuestions":[{"q":"Expand and simplify: -3(x - 4) + 2x","a":"-x + 12, because -3(x - 4) = -3x + 12, and -3x + 12 + 2x = -x + 12."},{"q":"A price p increases by 6% tax. Write the total as a single-term expression.","a":"1.06p, because p + 0.06p = 1p + 0.06p = 1.06p."},{"q":"Solve: -2/3 x - 5 = 1","a":"x = -9, because -2/3 x = 6, and dividing by -2/3 (multiplying by its reciprocal, -3/2) gives x = 6 x (-3/2) = -9."}]},"hero":"$undefined"}]

Linear expressions and equations with rational numbers