el":"A rollercoaster needs riders taller than 120 cm","note":"height > 120, an inequality with infinitely many solutions, not one exact height"},{"emoji":"💰","label":"You have at most £50 to spend","note":"total cost ≤ 50, a whole range of affordable combinations, not one fixed amount"},{"emoji":"🌡️","label":"Water is liquid between 0°C and 100°C","note":"a compound (two-sided) inequality: 0 < temperature < 100"},{"emoji":"🔄","label":"Solving -2x < 8 gives x > -4, not x < -4","note":"dividing by the negative -2 flips the inequality sign, the one rule that is easy to forget"}]},"sections":[{"id":"solving-inequalities","title":"1. Solving a linear inequality","phase":"Concrete","intro":"Solving an inequality uses the exact same moves as solving an equation: whatever you do to one side, do to the other, undoing operations in reverse order. The only new rule appears when you multiply or divide by a negative number.","body":["Solve 3x + 5 < 20. Subtract 5 from both sides: 3x < 15. Divide both sides by 3 (positive, so the inequality sign stays the same): x < 5.","Now solve -2x + 3 > 11. Subtract 3 from both sides: -2x > 8. Divide both sides by -2. Because -2 is NEGATIVE, the inequality sign flips from > to <: x < -4."],"worked":{"prompt":"Solve the inequality -2x + 3 > 11, and check the answer by substituting a value from the solution set back in.","steps":["Subtract 3 from both sides: -2x > 8.","Divide both sides by -2, flipping the sign because -2 is negative: x < -4.","Check with x = -5 (which satisfies x < -4): -2(-5) + 3 = 10 + 3 = 13, and 13 > 11 is true.","Check that a value OUTSIDE the solution set fails: x = -3 gives -2(-3) + 3 = 9, and 9 > 11 is false, as expected."],"answer":"x < -4"},"checks":["Why does dividing by a negative number flip the inequality sign, but dividing by a positive number does not?","How would you check your answer to an inequality, since there is no single value to substitute back in?"]},{"id":"number-line-representation","title":"2. Showing a solution set on a number line","phase":"Pictorial","intro":"A number line turns an inequality into a picture: a circle marks the boundary value, and shading shows every value in the solution set. The circle style tells you whether the boundary itself counts.","body":["For x > 3: draw an OPEN circle at 3 (since 3 itself is not greater than 3, it is not included), and shade to the RIGHT, toward larger numbers.","For x ≤ -1: draw a CLOSED (filled) circle at -1 (since -1 satisfies 'less than or equal to -1'), and shade to the LEFT, toward smaller numbers."],"figures":[{"kind":"numberLine","min":-2,"max":8,"step":1,"marks":[{"value":3,"label":"3","color":"#ffffff"}],"jumps":[{"from":3,"to":8,"label":"x > 3"}],"caption":"x > 3: an OPEN circle at 3 (white centre, so it is NOT filled in), shaded to the right toward larger values."}],"worked":{"prompt":"Show the compound inequality -2 ≤ x < 5 on a number line, and explain the circle at each end.","steps":["At x = -2: the inequality is ≤, so -2 IS included. Draw a CLOSED (filled) circle at -2.","At x = 5: the inequality is <, so 5 is NOT included. Draw an OPEN circle at 5.","Shade the whole region between -2 and 5, since every value strictly between them (and -2 itself) is a solution."],"answer":"A filled circle at -2, an open circle at 5, shaded between them."},"checks":["Why does -2 ≤ x < 5 need TWO different circle styles, one at each end?","Could a number line for a compound inequality ever have shading going in both directions, off both ends of the line?"]}],"guidedPractice":[{"question":"Solve 4x - 7 ≤ 9.","answer":"x ≤ 4, because 4x ≤ 16, then divide by the positive 4."},{"question":"Solve -3x + 1 < 10.","answer":"x > -3, because -3x < 9, then divide by -3 and flip the sign to get x > -3."},{"question":"Describe how x ≥ -2 would be shown on a number line.","answer":"A closed (filled) circle at -2, shaded to the right."},{"question":"A number line shows an open circle at 6, shaded to the left. Write the inequality.","answer":"x < 6"},{"question":"Write the inequality shown by a

Solving linear inequalities and number lines