Equations and inequalities

Linear equations

A linear equation has the variable only to the first power. Do the same thing to both sides until the variable is alone.

Example: \(3(x - 2) + 4 = 2x + 5\) becomes \(3x - 2 = 2x + 5\), so \(x = 7\). Check: \(3(7 - 2) + 4 = 19\) and \(2(7) + 5 = 19\).

All real numbers or no solution

If the variable disappears: a true statement means an identity (all real numbers); a false statement means a contradiction (no solution).

Example: \(2(x + 3) = 2x + 6\) gives \(6 = 6\): all real numbers. \(x + 4 = x - 1\) gives \(4 = -1\): no solution.

Equations with fractions or decimals

Multiply every term on both sides by the LCD (or by 10, 100) to get whole numbers.

Example: \(\dfrac{x}{3} + \dfrac{1}{2} = \dfrac{5}{6}\). Times 6: \(2x + 3 = 5\), so \(x = 1\).

Example: \(0.2x + 1.5 = 0.7\). Times 10: \(2x + 15 = 7\), so \(x = -4\).

Watch out: multiply every term, including the ones without fractions.

Formulas: solving for a letter

In a literal equation, treat the other letters like numbers and undo operations in reverse order.

Example: \(P = 2l + 2w\) for \(w\): \(w = \dfrac{P - 2l}{2}\). \(\;ax + b = c\) for \(x\): \(x = \dfrac{c - b}{a}\) (\(a \ne 0\)). \(\;y = mx + b\) for \(m\): \(m = \dfrac{y - b}{x}\) (\(x \ne 0\)).

Word problems: translating

"is" means \(=\). "of" with a percent means multiply. "5 less than \(x\)" is \(x - 5\) (order flips). Consecutive integers: \(n, n + 1, n + 2\); consecutive odd or even: \(n, n + 2, n + 4\). Distance: \(d = rt\).

Example: Three consecutive integers add to 48: \(n + (n + 1) + (n + 2) = 48\), so \(n = 15\): 15, 16, 17.

Example: A $80 jacket at 25% off costs \(0.75 \cdot 80 = 60\) dollars.

Linear inequalities

Solve like an equation, with one extra rule: when you multiply or divide both sides by a negative number, flip the inequality sign.

Example: \(3 - 2x \le 7\): \(-2x \le 4\), so \(x \ge -2\), which is \([-2, \infty)\).

Watch out: only flip for multiplying or dividing by a negative, not for subtracting.

Compound inequalities

A compound inequality with "and": both must be true (overlap). Solve a three-part one by doing the same step to all three parts. With "or": either is true; write a union \(\cup\).

Example: \(-3 \lt 2x + 1 \le 7\): \(-4 \lt 2x \le 6\), so \(-2 \lt x \le 3\), which is \((-2, 3]\).

Absolute value equations

First isolate the absolute value. Then for \(|X| = c\): if \(c \gt 0\), \(X = c\) or \(X = -c\); if \(c = 0\), \(X = 0\); if \(c \lt 0\), no solution.

Example: \(2|x - 3| + 1 = 9\): \(|x - 3| = 4\), so \(x - 3 = 4\) or \(x - 3 = -4\): \(x = 7\) or \(x = -1\).

Absolute value inequalities

"Less than" means between: \(|X| \lt c\) becomes \(-c \lt X \lt c\). "Greater than" means outside, "or": \(|X| \gt c\) becomes \(X \lt -c\) or \(X \gt c\). With \(\le\) or \(\ge\), the same, but the endpoints are included.

Example: \(|x - 2| \lt 5\): \(-3 \lt x \lt 7\), which is \((-3, 7)\).

Example: \(|2x + 1| \ge 3\): \(x \le -2\) or \(x \ge 1\), which is \((-\infty, -2] \cup [1, \infty)\).

Watch out: \(|X| \lt -2\) has no solution; \(|X| \gt -2\) is all real numbers.