Multi-Step Equations and Inequalities
Key steps, rules, and notation for solving multi-step linear equations and inequalities.
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Questions Covered in This Set
11 cards to master
What are the six steps of the universal recipe for solving a linear equation?
1) Distribute, 2) combine like terms on each side, 3) move variables to one side, 4) move constants to the other side, 5) divide by the coefficient, 6) check by substituting.
What is the most common mistake when distributing?
Distributing to only the first term — e.g. writing 5(x − 3) as 5x − 3 instead of 5x − 15. The factor must multiply EVERY term inside.
Simplify −2(x − 4).
−2x + 8 (negative times negative gives +8).
How do you handle an equation with fractions?
Multiply every term on both sides by the least common denominator (LCD) to clear the fractions.
You solve and get 0 = 0. What does that mean?
The equation is an identity — it's true for every x, so there are infinitely many solutions (the two lines are identical).
You solve and get 6 = 1. What does that mean?
No solution — the statement is never true (the lines are parallel and never meet).
When do you flip an inequality sign?
Only when you multiply or divide both sides by a negative number. Adding or subtracting never flips it.
Solve: −3x + 7 ≥ 22
−3x ≥ 15, then divide by −3 and FLIP: x ≤ −5.
How do you graph x ≤ −5 on a number line?
Closed (filled) circle at −5, shaded to the left. Open circles are used for strict < or >.
Write x ≤ −5 in interval notation, and explain the symbols.
(−∞, −5]. Brackets mean the endpoint is included, parentheses mean it isn't; infinity always gets a parenthesis.
Solve the compound inequality −1 < 2x + 3 ≤ 9.
Subtract 3 from all three parts: −4 < 2x ≤ 6; divide by 2: −2 < x ≤ 3, or (−2, 3].
