Study case: solving one modulus inequality visually and algebraically In this A-Level Modulus Inequality Graph Problem Tutor, I use one clear study case to explain how modulus inequalities work. The problem is: solve |2x - 3| < 7. The final answer is -2 < x < 5. I like this example because it can be solved in more than one way, and each way develops a different kind of understanding. Some students see modulus as a rule to memorise, but I want them to see it as distance and shape. Before solving, I ask the student what the modulus symbol means. The expression |2x - 3| represents the positive size of 2x - 3. If |2x - 3| < 7, then 2x - 3 is less than 7 units away from zero. That means it must lie between -7 and 7. This interpretation is the key to the first method. Method 1 is the compound inequality method. Since |2x - 3| < 7, I write -7 < 2x - 3 < 7. Then I solve the three-part inequality carefully. Add 3 to all three parts to get -4 < 2x < 10. Divide all three parts by 2 to get -2 < x < 5. This gives the solution interval. I emphasise that when the modulus is less than a positive number, the solution is usually between two boundary values. Method 2 is the two-boundary method. I find where the expression reaches the boundary values 7 and -7. First, 2x - 3 = 7 gives 2x = 10, so x = 5. Second, 2x - 3 = -7 gives 2x = -4, so x = -2. These are the two critical values. To decide the interval, I test a value between them, such as x = 0. Then |2(0) - 3| = 3, which is less than 7, so the region between -2 and 5 works. Values outside do not. Therefore -2 < x < 5. Method 3 is the graph method. I sketch y = |2x - 3| and the horizontal line y = 7. The modulus graph is V-shaped, and the line y = 7 crosses it at x = -2 and x = 5. The inequality asks where the modulus graph is below 7, so the solution is the interval between the two intersection points. This visual method is useful because it shows why the answer is an interval and not just two separate numbers. After the three methods, I compare them with the student. The compound inequality method is efficient. The boundary method is reliable when students are unsure about the inequality direction. The graph method gives the clearest understanding of the shape. I want students to know all three because modulus questions often change form. A question with greater than may give two outside regions instead of one middle interval. The most common mistakes are writing 2x - 3 < 7 only and forgetting the lower boundary, changing the inequality sign for no reason, giving x = -2 and x = 5 as the answer instead of an interval, and not checking whether the inequality is strict. Because the original inequality uses < rather than <=, the endpoints are not included. I ask students to pay close attention to open and closed boundaries. This topic links strongly to graphs. Modulus transformations appear in A-Level functions, solving equations and inequalities. If a student understands the V-shape of a modulus graph, many algebraic rules become easier to remember. I often ask students to sketch first, even if they later solve algebraically. The sketch does not need to be perfect; it only needs to show the critical points and the region required. A useful extension is to solve |2x - 3| > 7. The same boundary values appear, but the solution changes to x < -2 or x > 5 because the graph is above the line outside the intersection points. This contrast helps students understand the difference between less than and greater than modulus inequalities. In one-to-one tutoring, I would finish by asking the student to solve a similar inequality and explain which method they prefer. I would also ask them to draw a quick graph to confirm the interval. With practice, modulus inequalities become much less confusing because the student has both an algebraic method and a visual reason for the answer. For support with this topic, students can book one-to-one tutor services: https://www.mastermathstutoring.co.uk/services. For the wider course page, visit A-Level Maths tutoring: https://www.mastermathstutoring.co.uk/a-level-maths. To ask about lessons or availability, use contact MasterMaths Tutoring: https://www.mastermathstutoring.co.uk/contact-me.