Chapter 1
Equations and Inequalities
A year of algebra makes the routine feel settled: do the same thing to both sides and an answer appears. But a few of those familiar moves quietly change the question, whether it names an equation, an inequality with a direction to protect, or an absolute value with a fork built into its own definition. The statement you finish with is not always equivalent to the one you started with. Underneath every equation or inequality sits the collection of numbers that make it true, and the chapter turns on one question: which moves leave that collection exactly as it was?
What You'll Explore
4 lessons.
- Solving Linear Equations
Solving has always ended with a number. What does it mean, then, for an equation to have no answer at all, or for every number at once to be an answer? This lesson goes back to what an equation asks of a number, and to what a step is really supposed to protect.
- Literal Equations and Formulas
Rearranging a formula to isolate a different letter is familiar work. What is different here is that you can no longer inspect the quantity you are dividing by, since it is a letter and nobody told you its value. This lesson asks what an honest answer looks like under that much uncertainty.
- Linear Inequalities
Once an equation becomes an inequality, solving stops asking which numbers make two sides equal and starts asking which numbers make one side bigger. That comparison usually returns a whole region of the number line instead of isolated points, and one of your usual moves stops working safely once order enters the picture. This lesson finds out why.
- Absolute Value Equations and Inequalities
Distance from zero never carries a sign. So if you are told only how far a number sits from zero, do you know which number it is? Everything built on absolute value inherits whatever the answer to that turns out to be, and this lesson follows it into equations and inequalities alike.