Equations, Inequalities & Systems

1 · Multi-step equations

Distribute, collect like terms, then undo one step at a time. 3(2x − 5) = 4x + 7 becomes 6x − 15 = 4x + 7, then 2x = 22 and x = 11. Check: 3(17) = 51 and 44 + 7 = 51.

2 · Equations with fractions

Multiply every term by a common denominator to clear the fractions. x/3 + x/4 = 14: multiply by 12 to get 4x + 3x = 168, so x = 24.

With one fraction on each side, cross-multiply and keep the parentheses: (x + 2)/5 = (x − 1)/3 gives 3(x + 2) = 5(x − 1), so 3x + 6 = 5x − 5 and x = 5.5.

3 · Solving a formula for one letter

Treat the other letters like numbers and undo the steps around the one you want. A = ½bh: multiply by 2 to get 2A = bh, then divide by b: h = 2A/b.

v = u + at: subtract u, then divide by a: t = (v − u)/a. Writing v − u/a divides only the u.

4 · Inequalities

Solve like an equation, with one extra rule: multiplying or dividing by a negative number flips the sign. 5 − 2x > 11 → −2x > 6 → x < −3.

5 · Compound inequalities

Do the same thing to all three parts. −3 < 2x + 1 ≤ 9: subtract 1 to get −4 < 2x ≤ 8, divide by 2 to get −2 < x ≤ 4. The integers that work are −1, 0, 1, 2, 3, 4: six of them. Watch which ends are included.

6 · Absolute value

|x − 4| is the distance from x to 4. |x − 4| = 7 means x is 7 away from 4: x = 11 or x = −3.

|x − 4| < 7 means closer than 7: −3 < x < 11. |x − 4| > 7 means farther: x < −3 or x > 11.

7 · Systems by substitution

When one equation already says what a letter equals, put that into the other one. y = 2x − 1 and 3x + y = 14: 3x + (2x − 1) = 14, so 5x = 15, x = 3, and y = 5.

8 · Systems by elimination

Add or subtract the equations so one letter cancels. 2x + 3y = 12 and 2x − y = 4: subtracting gives 4y = 8, so y = 2, and then 2x − 2 = 4 gives x = 3.

Sometimes you need to multiply one equation first so the numbers in front of a letter match.

9 · Systems from word problems

A play sold 100 tickets for $650; adult tickets cost $8 and student tickets $5. If all 100 were student tickets, they would bring $500. Each adult ticket adds $3, and the extra $150 ÷ 3 = 50 adult tickets.

The same reasoning works for coins, animals with different numbers of legs, and any "two kinds, one total" problem.

10 · No solution, or every solution

Two equations are two lines. Parallel lines never meet: 2x + y = 5 and 4x + 2y = 7 have the same slope but different intercepts, so there is no solution.

If one equation is just a multiple of the other (2x + y = 5 and 4x + 2y = 10), they are the same line, and there are infinitely many solutions.