Introduction to Algebra: Chapter Review

Vocabulary and notation

Variable

A letter standing for a number that is unknown or free to change. A number written against a letter means multiply, so $3n$ is $3 \times n$.

Term

One of the pieces an expression splits into at its $+$ and $-$ signs, carrying the sign in front of it: $5x + 3y - 7$ has terms $5x$, $3y$, and $-7$.

Coefficient

The number multiplying the variable in a term. Plain $x$ means $1x$ and $-x$ means $-1x$, so $4x - x = 3x$, not $4$.

Constant

A term with no variable at all, so its value never changes. In $9k + 2$ the constant is $2$ and the coefficient is $9$.

Like terms

Terms whose variable parts are identical: same letters, same powers. $3x$ and $5x$ are like; $3x$ and $3x^2$ are not, nor are $4x$ and $4y$. Two constants are always like.

Expression

A phrase with no relation symbol, such as $2a - 7$: it names a value, so there is nothing to solve. You evaluate it (substitute values, reach a number) or simplify it (no values needed, reach a shorter equivalent).

Equation

Two expressions joined by $=$, a sentence that can be true or false, such as $3n = 12$. Its solution is the variable value making both sides equal; every equation in this chapter has exactly one.

Inequality

Two expressions joined by $<$, $>$, $\le$, or $\ge$. Its solution is every value making it true, almost always an infinite range, so the answer is a ray, not a point.

Inverse operations

The pairs that undo each other: addition with subtraction, multiplication with division. Isolating a variable means applying the inverse of what is attached to it, to both sides.

Strict and non-strict symbols

$<$ and $>$ are strict and exclude the boundary, so $3 > 3$ is false. $\le$ (at most) and $\ge$ (at least) are non-strict and include it, so $3 \ge 3$ is true.

Formulas and theorems

• Translating words into symbols

  $$\begin{gathered} \text{5 less than } x \;\to\; x - 5 \\ \text{twice the sum of } x \text{ and } 4 \;\to\; 2(x + 4) \end{gathered}$$

  Use when "Less than", "fewer than", and "subtracted from" reverse the spoken order. Comparison words fix the symbol: at most $\le$, at least $\ge$, more than $>$; "less than" gives $<$ when it links two quantities ($x$ is less than $5$) but subtracts when it names an amount ($5$ less than $x$).

• Evaluating by substitution

  Replace every occurrence of each variable with its value in parentheses, then run the full order of operations.

  Use when Needs a value for every variable. The parentheses protect the sign and the coefficient: $(-4)^2 = 16$ but $-4^2 = -16$, and an exponent on the variable reaches only the substituted value, never the coefficient.

  e.g. $2x^2 - 4x - 1$ at $x = -3$: $2(9) + 12 - 1 = 29$.

• Combining like terms

  $$ma + na = (m + n)a$$

  Use when Legal only when the variable parts are identical, same letters and same exponents. Add the coefficients only; exponents are never added.

  e.g. $5x^2 - 2x + x^2 + 6x = 6x^2 + 4x$.

• Distributive property

  $$\begin{gathered} a(b + c) = ab + ac \\ a(b - c) = ab - ac \end{gathered}$$

  Use when Any numbers or terms. The outside factor multiplies every term inside; a leading minus is the factor $-1$, flipping the sign of each term.

  e.g. $\tfrac{1}{3}(9x - 12) = 3x - 4$.

• Properties of equality (the balance principle)

  $$\begin{gathered} \text{if } a = b, \text{ then} \\ a + c = b + c, \quad a - c = b - c \\ ac = bc, \quad \dfrac{a}{c} = \dfrac{b}{c} \end{gathered}$$

  Use when The same $c$ must hit the whole of both sides, and $c \neq 0$ for both the multiplication and the division: multiplying by $0$ gives the useless $0 = 0$.

• The four one-step forms

  $$\begin{gathered} x + a = b \;\Rightarrow\; x = b - a \\ x - a = b \;\Rightarrow\; x = b + a \\ ax = b \;\Rightarrow\; x = \dfrac{b}{a} \\ \dfrac{x}{a} = b \;\Rightarrow\; x = ab \end{gathered}$$

  Use when $a \neq 0$ in the multiplication and division forms. A fractional coefficient is undone by multiplying by its reciprocal.

• The two-step forms

  $$\begin{gathered} ax + b = c \;\Rightarrow\; x = \dfrac{c - b}{a} \\ \dfrac{x}{a} + b = c \;\Rightarrow\; x = a(c - b) \end{gathered}$$

  Use when $a \neq 0$. A subtracted constant is cleared by adding it, so $ax - b = c$ gives $x = \dfrac{c + b}{a}$. Undo in reverse of the order of operations: constant first, coefficient second. Dividing first is legal and reaches the same answer, but it splits $b$ into a fraction.

  e.g. $\dfrac{x}{2} - 4 = -1$: $x = 2(-1 + 4) = 6$.

• Adding or subtracting across an inequality

  $$a < b \;\Rightarrow\; a + c < b + c$$

  Use when Any $c$, and likewise for $>$, $\le$, $\ge$. The symbol never changes at an add-or-subtract step, even when the number added is negative.

• Multiplying or dividing an inequality by a positive

  $$\begin{gathered} a < b \text{ and } c > 0 \\ \Rightarrow\; ac < bc \;\text{ and }\; \dfrac{a}{c} < \dfrac{b}{c} \end{gathered}$$

  Use when $c$ strictly positive. The symbol is unchanged, and the same holds for $>$, $\le$, $\ge$.

• Multiplying or dividing an inequality by a negative

  $$\begin{gathered} a < b \text{ and } c < 0 \\ \Rightarrow\; ac > bc \;\text{ and }\; \dfrac{a}{c} > \dfrac{b}{c} \end{gathered}$$

  Use when $c$ strictly negative, so $<$ becomes $>$ and $\le$ becomes $\ge$. The sign of $c$ must be known, so never multiply or divide both sides by a variable. It flips once per multiply-or-divide step.

  e.g. $2 < 3$ multiplied by $-1$ gives $-2 > -3$.

• Graphing a solution range

  $$\begin{gathered} x > a, \; x \ge a: \text{ shade right} \\ x < a, \; x \le a: \text{ shade left} \end{gathered}$$

  Text description

  Two number lines at the same boundary a: the upper has an open circle and shades right for x greater than a, the lower a closed circle and shades left for x less than or equal to a.

  Use when Isolate the variable on the left first: $4 < x$ is $x > 4$, so it shades right. Open circle on $a$ for the strict $<$ and $>$, closed for $\le$ and $\ge$; the arrowhead says the range runs on forever.

Problem types, step by step

Exam traps

• Trap Dividing $-2x < 8$ by $-2$ and carrying the symbol straight down, giving $x < -4$.

  Fix Dividing by a negative reverses it, so $x > -4$. Test $x = 0$ in the original: $0 < 8$ is true, yet $0$ is nowhere in $x < -4$.

• Trap Flipping the symbol because a negative appears somewhere, such as subtracting $1$ from both sides of $-2x + 1 > 9$.

  Fix Only multiplying or dividing both sides by a negative flips it: that step gives $-2x > 8$ with $>$ intact, and the flip arrives at the division by $-2$, giving $x < -4$.

• Trap Changing only one side, turning $2x + 3 = 11$ into $2x = 11$.

  Fix Subtract $3$ from both sides for $2x = 8$ and $x = 4$. The one-sided version gives $x = 5.5$, and $2(5.5) + 3 = 14 \neq 11$.

• Trap Subtracting the coefficient instead of dividing, turning $6x = 42$ into $x = 36$.

  Fix $6x$ means $6 \cdot x$, not $6 + x$, so divide both sides by $6$: $x = 7$.

• Trap Dividing $-3x = -6$ by a bare $3$ and writing $x = -2$.

  Fix Divide by the full coefficient $-3$, sign included: $x = 2$, and $-3(2) = -6$ confirms it.

• Trap Letting a leading minus reach only the first term, so $9 - (x + 4)$ becomes $9 - x + 4$.

  Fix The minus is a factor of $-1$ on every term inside: $9 - x - 4 = 5 - x$.

• Trap Adding the exponents or the letters when combining, writing $2x + 3x = 5x^2$ or $3x + 4y = 7xy$.

  Fix Add coefficients only and keep the variable part: $2x + 3x = 5x$. Unlike terms share no factor to collect, so $3x + 4y$ is already simplest.

• Trap Reading "five less than $x$" left to right as $5 - x$.

  Fix It is $x - 5$: start at $x$ and take five away. At $x = 2$ the two differ, $-3$ against $3$.

• Trap Dropping the parentheses around a negative input, so $-x^2$ at $x = -6$ comes out as $36$.

  Fix $-x^2$ squares the value first and negates after: $-(-6)^2 = -36$. Only $(-6)^2$, with the sign inside, is $36$.