Solving equations and inequalities Algebra 1 — 8-topic bundle

An equation is a mathematical statement that shows two expressions are equal. It always has an equals sign (=).

For example: $x + 5 = 10$

Here, $x+5$ is one expression and $10$ is another. The = sign says they're the same value. Our job is to find what $x$ has to be for that statement to be true.

When we talk about inequalities, we're dealing with relationships between numbers that aren't strictly equal. Instead, one quantity might be greater than, less than, greater than or equal to, or less than or equal to another.

Here are the basic inequality symbols:
* $ < $ : less than (e.g., $ 3 < 5 $)
* $ > $ : greater than (e.g., $ 7 > 2 $)
* $ \le $ : less than or equal to (e.g., $ 4 \le 4 $, $ 4 \le 6 $)
* $ \ge $ : greater than or equal to (e.g., $ 9 \ge 9 $, $ 9 \ge 5 $)
* $ \ne $ : not equal to (e.g., $ 5 \ne 8 $)

When you see an absolute value equation like |x| = 5, it's asking: "What number(s) are 5 units away from zero?" The answers are x = 5 and x = -5.

The key to solving these equations is to remember that the expression inside the absolute value bars could be either positive or negative to get the final positive result.

When you're solving a linear inequality, you're looking for all the values of the variable that make the statement true. It's very similar to solving equations, with just one crucial rule to remember.

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When you're solving linear equations with one variable, you're looking for the single value that makes the equation true. The key is to isolate the variable, meaning you want to get it by itself on one side of the equals sign. You achieve this by performing inverse operations.

Compound inequalities link two simple inequalities using either "and" or "or." How you combine their solutions depends entirely on which word is used.

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