An equation finds "exactly what value," while an inequality finds "which range." For example, 2x + 3 = 11 is only true when x is exactly 4, but 2x + 3 > 11 is true for every number greater than 4. Solving one works almost exactly like solving the other, except for one thing: multiplying or dividing by a negative number.
Let's check why with numbers. 2 < 5 is true. What happens if you multiply both sides by −1? You get −2 and −5, and −2 < −5 turns out to be false. What's actually true is −2 > −5. So the moment you multiply both sides by a negative number, the inequality sign has to flip on its own for the statement to stay true. This isn't a rule you memorize — it's a natural consequence of the fact that multiplying by a negative number swaps the positions of the larger and smaller number on the number line.
So when solving a linear inequality like ax + b > c, subtracting b from both sides works exactly like it does for an equation (ax > c−b). The tricky part comes at the very end, when you divide both sides by a. If a is positive, keep the direction the same; if a is negative, you have to flip it. Miss this one step and the range of your answer comes out completely backwards. That's why, when solving inequality problems, it's important to always double-check at the end: "is the number I'm dividing by negative?"
Now let's move on to systems of equations. A system of equations is a problem where you find the values that satisfy two equations with two unknowns (usually x and y) at the same time. For example, you'd need to find the x and y that satisfy both x + y = 5 and x − y = 1 at once. The method used here is called "elimination" — adding or subtracting the two equations to make one of the unknowns disappear entirely. In the example above, adding the two equations makes y vanish: (x+y)+(x−y) = 5+1 → 2x = 6 → x = 3. Once you have x, substitute it into either equation to get y = 2.
Seeing this on a graph makes it much more intuitive. x + y = 5 and x − y = 1 each become a single line on the coordinate plane. The two lines usually meet at one point, and that intersection point is exactly the (x, y) value that satisfies both equations at once. So "solving a system of equations" ultimately means the same thing as "finding where two lines meet." Checking it on a graph even lets you verify your calculation with your own eyes.
Inequalities and systems of equations might look like different concepts, but they both stand on the same principle — using the properties of an equation (or inequality) to "do the same operation on both sides." Carry over the feel you built up in the Equation Balance Lab, and both concepts click into place much more naturally.
On our activity page, you can solve an inequality step by step and see the direction actually flip when you divide by a negative number, and for the system of equations, you can graph the two lines and try to guess the intersection point yourself.