Equations and Inequalities: Chapter Review

Vocabulary and notation

Solution set

The set of reals making a statement true; $\mathbb{R}$ is all reals, $\varnothing$ none. Solving describes that set, not just one value.

Conditional, identity, contradiction

True for some values but not all; true for every real; true for none. Every equation is exactly one of the three.

Reversible step

A step you can undo. It guarantees an EQUIVALENT equation, one with the same solution set, but guarantees nothing for an inequality: $x < 3$ times $-1$ is reversible, yet $-x < -3$ names a different set.

Extraneous and lost solutions

A candidate solving a derived statement but not the original is extraneous; a solution the original had and the derived one dropped is lost.

Excluded value

A value making a denominator of the ORIGINAL zero, so never a solution whatever the algebra says later.

Unknown and parameter

In a literal equation (more than one letter), the letter solved for is the unknown, each letter held fixed a parameter; the question assigns the roles, not the alphabet.

Compound inequality

Two conditions joined by "and", the intersection of their solution sets, or by "or", the union; a chain $a < x < b$ is an "and".

Breakpoint

A value where an expression inside bars changes sign; breakpoints cut the line into intervals on which every bar resolves to one formula.

Formulas and theorems

• The trichotomy of $ax = b$

  $$\begin{gathered} a \neq 0:\ x = \frac{b}{a} \\ a = 0,\ b \neq 0:\ \varnothing \\ a = 0,\ b = 0:\ \mathbb{R} \end{gathered}$$

  Use when Any reals $a$, $b$, numbers or parameters. Exhaustive, so a linear equation never has exactly two solutions; a parameter $a$ hides which case holds, so carry all three.

• Safe moves on an equation

  Add any expression defined for every $x$ to both sides; multiply or divide both sides by a nonzero constant.

  Use when An expression that can vanish ADDS roots at its zeros when you multiply and LOSES them when you divide; multiplying by $0$ destroys the equation.

• Solving for a letter in several terms

  $$\begin{gathered} ax + b = cx + d \\ \Longrightarrow\ (a - c)x = d - b \\ \Longrightarrow\ x = \frac{d - b}{a - c} \end{gathered}$$

  Use when $a \neq c$, a condition on the whole factor divided by, not its letters: $a = c = 5$ fails though neither is zero. At $a = c$ the trichotomy takes over.

• Where a rearranged formula gains a condition

  A rearrangement gains a NEW restriction exactly where it divides by an expression that can be zero, and none where it divides only by constants.

  Use when Scope: rearrangements built from adding constants and multiplying or dividing; the original's own restrictions still apply.

  e.g. $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ gives $v = \frac{fu}{u - f}$, adding $u \neq f$.

• Order rules for inequalities

  $$\begin{gathered} c > 0:\ a < b \iff ca < cb \\ c < 0:\ a < b \iff ca > cb \end{gathered}$$

  Use when Adding any expression defined for every $x$ keeps set and direction. Only the sign of what multiplies BOTH sides flips the symbol; multiplying by $0$ destroys the inequality; an unknown-sign expression is a fork, not a step.

  e.g. $-3x > 12$ gives $x < -4$; at $x = -5$, $15 > 12$.

• The four cases of $ax > b$

  $$\begin{gathered} a > 0:\ x > \tfrac{b}{a} \\ a < 0:\ x < \tfrac{b}{a} \\ a = 0,\ b < 0:\ \mathbb{R} \\ a = 0,\ b \ge 0:\ \varnothing \end{gathered}$$

  Use when Any reals $a$, $b$. The strict boundary falls opposite to the equation's: at $a = b = 0$ this is $\varnothing$, since $0 > 0$ is false, while $ax = b$ is $\mathbb{R}$. For $ax \ge b$ the $\mathbb{R}$ branch becomes $b \le 0$.

• Absolute value, defined and squared

  $$\begin{gathered} \lvert x \rvert = \begin{cases} x & \text{if } x \ge 0, \\ -x & \text{if } x < 0, \end{cases} \\ \lvert x \rvert = \sqrt{x^2} \end{gathered}$$

  Use when Both hold for every real $x$. The bars are a hidden if, so a bar holding an unknown carries an unresolved case.

• $\lvert X \rvert = c$ for a constant $c$

  $$\begin{gathered} c > 0:\ X = c \ \text{ or } \ X = -c \\ c = 0:\ X = 0 \\ c < 0:\ \varnothing \end{gathered}$$

  Use when Check the sign of $c$ before splitting: only $c > 0$ gives two cases.

• $\lvert X \rvert = Y$ with an unknown on the right

  $$\begin{gathered} \lvert X \rvert = Y \iff \\ Y \ge 0 \ \text{ and } \\ \bigl(X = Y \ \text{ or } \ X = -Y\bigr) \end{gathered}$$

  Use when Any expressions $X$, $Y$. The bare split drops $Y \ge 0$ and runs one way only, so test each candidate against $Y \ge 0$; both can fail, leaving $\varnothing$.

• $\lvert X \rvert = \lvert Y \rvert$

  $$\begin{gathered} \lvert X \rvert = \lvert Y \rvert \\ \iff\ X = Y \ \text{ or } \ X = -Y \\ \iff\ X^2 = Y^2 \end{gathered}$$

  Use when An equivalence for any $X$, $Y$: no side condition, since both sides are nonnegative, so nothing is discarded and squaring is exact.

  e.g. $\lvert x - 6 \rvert = \lvert x + 2 \rvert$: one case gives $-6 = 2$, so only $x = 2$.

• Absolute-value inequalities

  $$\begin{gathered} \lvert X \rvert < c \iff -c < X < c \\ \lvert X \rvert > c \iff X < -c \ \text{ or } \ X > c \end{gathered}$$

  Text description

  A number line on which the absolute value of X is less than c, giving one band between minus c and c, above a second number line on which the absolute value of X is greater than c, giving two rays running outward from those same two points.

  Use when Both hold for EVERY real $c$: the band is nonempty only when $c > 0$, since at $c \le 0$ its ends cross, and the rays cover $\mathbb{R}$ when $c < 0$ and say $X \neq 0$ at $c = 0$. Also $\lvert X \rvert \le 0$ forces $X = 0$, and $\lvert x - a \rvert \le c$ is $[a - c, a + c]$.

  e.g. $\lvert 3x - 5 \rvert < -2$ is $\varnothing$; $\lvert 4 - 7x \rvert > -1$ is $\mathbb{R}$.

• Sum of two distances

  $$\begin{gathered} \lvert x - p \rvert + \lvert x - q \rvert = d \\ g = \lvert p - q \rvert \end{gathered}$$

  Text description

  The case where d is greater than g, on a number line: one solution sits outside each anchor, both the same step of d minus g over 2 beyond it, which is why no solution exists once d is less than g.

  Use when Anchors $p$, $q$, any real $d$: $d < g$ gives $\varnothing$, $d = g$ the closed interval between the anchors, and $d > g$ two solutions, one step of $\frac{d - g}{2}$ outside each anchor.

  e.g. $\lvert x - 6 \rvert + \lvert x + 2 \rvert = 12$: $g = 8$, so $x = -4$ or $x = 8$.

Problem types, step by step

Exam traps

• Trap Dividing both sides by a variable: from $x^2 = x$, dividing by $x$ reports $\{1\}$.

  Fix That destroys $x = 0$. Factor instead: $x(x - 1) = 0$ gives $\{0, 1\}$.

• Trap Confirming a candidate in the CLEARED equation, so $\frac{x + 3}{x - 4} = \frac{7}{x - 4}$ is answered $x = 4$.

  Fix $x = 4$ is the excluded value, so it is extraneous and the solution set is $\varnothing$.

• Trap Reversing the inequality symbol because a negative number appears somewhere in the problem.

  Fix Only the sign of what multiplies or divides BOTH sides flips it; a negative constant elsewhere, or a subtraction, never does.

• Trap Multiplying an inequality by the variable to clear a fraction, so $\frac{1}{x} < 1$ is answered $x > 1$.

  Fix That treats $x$ as positive and loses the ray $x < 0$. Splitting on the sign gives $x < 0$ or $x > 1$.

• Trap Reporting both cases of $\lvert X \rvert = Y$, so $\lvert 2x + 1 \rvert = x - 3$ is answered $-4$ and $\frac{2}{3}$.

  Fix Both fail $x - 3 \ge 0$, so both are extraneous and the solution set is $\varnothing$.

• Trap Simplifying $\sqrt{x^2}$ to $x$.

  Fix It equals $\lvert x \rvert$: at $x = -5$ the value is $5$, since a square root is never negative.

• Trap Treating $\lvert X \rvert > 0$ as true for every real number.

  Fix It holds everywhere except where $X = 0$; the one true for every real number is $\lvert X \rvert \ge 0$.