If you've already learned quadratic equations and quadratic inequalities, this time we'll look at what happens when several of them overlap, and at equations and inequalities that involve absolute value. All three look different at first glance, but they share the same underlying idea: finding the x that satisfies several conditions at the same time.
Let's start with systems of quadratic equations. To find the points that satisfy both the line y=mx+k and the parabola y=x² at once, you set the two expressions equal — x²=mx+k, or x²−mx−k=0 — and solve that quadratic equation. Looking at its discriminant D=m²+4k: if D>0 the line and parabola meet at two points, if D=0 they touch at exactly one point, and if D<0 they never meet at all. In other words, a problem about where two graphs meet turns into a problem about the discriminant of a quadratic equation.
A system of inequalities asks you to find every x that satisfies two or more inequalities at the same time. If you draw each inequality's solution on a number line, some stretches satisfy only one inequality, while others satisfy both. The solution to the system is exactly that overlapping stretch where both are satisfied — the same idea as finding people who meet two conditions at once, where you take the intersection of the two conditions.
Finally, equations and inequalities with absolute value. The absolute value |x−a| really just means "the distance between x and a on the number line." So |x−a|=b means "find the points that are exactly a distance of b from a," and the answer is always the two points a−b and a+b, symmetric around a. |x−a|<b describes "the range that's closer to a than b," so the whole stretch between a−b and a+b is the solution; conversely |x−a|>b describes "the range that's farther from a than b," so everything outside that stretch is the solution. Rather than mechanically splitting the absolute value into two cases, picturing it as "distance" first makes it far less confusing.
All three of these show up often in real life. A system of quadratic equations is exactly what you need when you're looking for a point that satisfies two conditions at once (say, a budget and a minimum size), while absolute-value inequalities are perfect for situations where something must "fall within a margin of error around a target value" — for example, a part's length must be within ±0.1 cm of 10 cm, i.e. |length−10|≤0.1. On our activity page, the three ideas are split into tabs, and you can change the values with sliders to see the intersection points on a graph, the overlapping range on a number line, and the distance concept behind absolute value, all for yourself.