Math by First Principles
Chapter Review · a rapid pre-test review (speedrun)

Introduction to Algebra: Chapter Review

A rapid review before the test: the chapter's vocabulary and notation, every formula with the conditions to use it, the standard problem types step by step, and the traps that cost points.

Vocabulary and notation

Variable
A letter standing for a number that is unknown or free to change. A number written next to a letter means multiply, so 3n3n is 3×n3 \times n.
Term
One of the pieces an expression splits into at its main-level ++ and −- signs (not signs inside parentheses), carrying the sign in front of it: 5x+3y−75x + 3y - 7 has terms 5x5x, 3y3y, and −7-7.
Coefficient
The number multiplying the variable in a term. Plain xx means 1x1x and −x-x means −1x-1x, so 4x−x=3x4x - x = 3x, not 44.
Constant
A term with no variable at all, so its value never changes. In 9k+29k + 2 the constant is 22 and the coefficient is 99.
Like terms
Terms whose variable parts are identical: same letters, same powers. 3x3x and 5x5x are like; 3x3x and 3x23x^2 are not, nor are 4x4x and 4y4y. Two constants are always like.
Expression
A phrase with no relation symbol, such as 2a−72a - 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=123n = 12. Its solution is any variable value making both sides equal. The equations solved in the one-step and two-step lessons are each built to have 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>33 > 3 is false. ≤\le (at most) and ≥\ge (at least) are non-strict and include it, so 3≥33 \ge 3 is true.

Formulas and theorems

  • Translating words into symbols

    5 less than x  →  x−5twice the sum of x and 4  →  2(x+4)\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" and "less than" each work two ways: they add or subtract when naming an amount ("seven more than xx" is x+7x + 7; "five less than xx" is x−5x - 5), but they compare when linking two quantities ("xx is more than 77" is x>7x > 7; "xx is less than 77" is x<7x < 7).

  • 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(-4)^2 = 16 but −42=−16-4^2 = -16, and an exponent on the variable reaches only the substituted value, never the coefficient.

    e.g. 2x2−4x−12x^2 - 4x - 1 at x=−3x = -3: 2(9)+12−1=292(9) + 12 - 1 = 29.

  • Combining like terms

    ma+na=(m+n)ama + 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. 5x2−2x+x2+6x=6x2+4x5x^2 - 2x + x^2 + 6x = 6x^2 + 4x.

  • Distributive property

    a(b+c)=ab+aca(b−c)=ab−ac\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 −1-1, flipping the sign of each term.

    e.g. 13(9x−12)=3x−4\tfrac{1}{3}(9x - 12) = 3x - 4.

  • Properties of equality (the balance principle)

    if a=b, thena+c=b+c,a−c=b−cac=bc,ac=bc\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 cc must hit the whole of both sides. Division needs c≠0c \neq 0. Multiplying by c=0c = 0 still keeps both sides equal, but it collapses them to the useless 0=00 = 0 and destroys the original solution, so avoid it too when solving.

  • The four one-step forms

    x+a=b  ⇒  x=b−ax−a=b  ⇒  x=b+aax=b  ⇒  x=baxa=b  ⇒  x=ab\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≠0a \neq 0 in the multiplication and division forms. A fractional coefficient is undone by multiplying by its reciprocal.

  • The two-step forms

    ax+b=c  ⇒  x=c−baxa+b=c  ⇒  x=a(c−b)\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≠0a \neq 0. A subtracted constant is cleared by adding it, so ax−b=cax - b = c gives x=c+bax = \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 bb into a fraction.

    e.g. x2−4=−1\dfrac{x}{2} - 4 = -1: x=2(−1+4)=6x = 2(-1 + 4) = 6.

  • Adding or subtracting across an inequality

    a<b  ⇒  a+c<b+ca < b \;\Rightarrow\; a + c < b + c

    Use when Any cc, 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

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

    Use when cc strictly positive. The symbol is unchanged, and the same holds for >>, ≤\le, ≥\ge.

  • Multiplying or dividing an inequality by a negative

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

    Use when cc strictly negative, so << becomes >> and ≤\le becomes ≥\ge. The sign of cc must be known before deciding whether to flip, which is why every multiply-or-divide step in this chapter uses a known number, not a variable. It flips once per multiply-or-divide step.

    e.g. 2<32 < 3 multiplied by −1-1 gives −2>−3-2 > -3.

  • Graphing a solution range

    x>a,  x≥a: shade rightx<a,  x≤a: shade left\begin{gathered} x > a, \; x \ge a: \text{ shade right} \\ x < a, \; x \le a: \text{ shade left} \end{gathered}
    An open circle leaves the boundary out, a closed circle takes it inTwo horizontal number lines, each with a tick either side of a boundary point labeled a at the center. The upper line carries a hollow circle at a with a thick shaded ray running right from the circle to an arrowhead, labeled x greater than a. The lower line carries a solid circle at a with a thick shaded ray running left to an arrowhead, labeled x less than or equal to a.x > aopen: a is outax ≤ aclosed: a is ina
    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<x4 < x is x>4x > 4, so it shades right. Open circle on aa for the strict << and >>, closed for ≤\le and ≥\ge; the arrowhead says the range runs on forever.

Problem types, step by step

Translate words into an expression, equation, or inequality

  1. Name the unknown with a letter.
  2. Convert each phrase to its operation, reversing the order for "less than" and "subtracted from".
  3. Wrap in parentheses any sum or difference that a later operation acts on as a whole.
  4. For a comparison, pick the symbol from the comparison words and check whether the boundary is allowed.
  5. An equation or inequality still needs solving; a bare expression has nothing to solve, only to evaluate or simplify. State the answer in the problem's own terms.

e.g. A 55 dollar fee plus 22 dollars a mile, at most 1919 dollars: 5+2m≤195 + 2m \le 19, so m≤7m \le 7.

Evaluate an expression at given values

  1. Substitute every occurrence of each variable, each value in its own parentheses.
  2. Finish grouped pieces first, then exponents, then multiplication and division left to right, then addition and subtraction.

e.g. x2+3xx^2 + 3x at x=−4x = -4: (−4)2+3(−4)=16−12=4(-4)^2 + 3(-4) = 16 - 12 = 4.

Simplify an expression

  1. Distribute across every set of parentheses, reading a leading minus as the factor −1-1.
  2. Sort the terms into groups with identical variable parts, each keeping the sign in front of it.
  3. Add the coefficients within each group and leave the variable part unchanged.
  4. Leave unlike groups separate; simplest means no two remaining terms are like.

e.g. 4(2x−3)−3(x−5)=8x−12−3x+15=5x+34(2x - 3) - 3(x - 5) = 8x - 12 - 3x + 15 = 5x + 3.

Solve a one-step equation

  1. Read which single operation is attached to the variable.
  2. Apply its inverse to both sides, carrying any negative sign with the whole coefficient.
  3. Simplify each side so the variable stands alone.

e.g. −34x=9-\tfrac{3}{4}x = 9: multiply both sides by −43-\tfrac{4}{3}, giving x=−12x = -12.

Solve a two-step equation

  1. Simplify each side first: distribute, then combine like terms.
  2. Add or subtract to clear the constant, leaving the variable term alone.
  3. Divide by the coefficient, or multiply by the divisor, to isolate the variable.
  4. Check in the original equation, before any simplifying.

e.g. 3x+2x+4=193x + 2x + 4 = 19: 5x+4=195x + 4 = 19, then 5x=155x = 15, so x=3x = 3.

Solve an inequality

  1. Run the same sequence as for an equation, copying the symbol down each line.
  2. At every multiply-or-divide step, reverse the symbol if the number used is negative.
  3. Never reverse at an add-or-subtract step, whatever signs appear elsewhere.
  4. If the variable ends up on the right, read the statement from the other end.

e.g. 8−3x≥28 - 3x \ge 2: −3x≥−6-3x \ge -6, and dividing by −3-3 flips it to x≤2x \le 2.

Graph an inequality on a number line

  1. Solve until the variable stands alone on the left.
  2. Mark the boundary number, with an open circle for << or >> and a closed circle for ≤\le or ≥\ge.
  3. Shade toward the larger numbers for >> or ≥\ge and toward the smaller for << or ≤\le.

e.g. 2x+1≥72x + 1 \ge 7 gives x≥3x \ge 3: closed circle on 33, shaded right.

Check an answer

  1. For an equation, substitute into the original and confirm the two sides are the same number.
  2. For an inequality, test one value from inside the claimed range (the original must come out true) and one from outside (it must come out false).
  3. For ≤\le or ≥\ge, test the boundary itself, which must make the original true.
  4. If a test disagrees, suspect a missed sign flip or a step applied to only one side.
  5. A few passing tests catch an error but do not prove the whole range is correct; only the algebra does that.

e.g. x≥4x \ge 4 from x2−3≥−1\dfrac{x}{2} - 3 \ge -1: x=6x = 6 gives 0≥−10 \ge -1, true, and x=0x = 0 gives −3≥−1-3 \ge -1, false.

Exam traps

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

    Fix Dividing by a negative reverses it, so x>−4x > -4. Test x=0x = 0 in the original: 0<80 < 8 is true, yet 00 is nowhere in x<−4x < -4.

  • Trap Flipping the symbol because a negative appears somewhere, such as subtracting 11 from both sides of −2x+1>9-2x + 1 > 9.

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

  • Trap Changing only one side, turning 2x+3=112x + 3 = 11 into 2x=112x = 11.

    Fix Subtract 33 from both sides for 2x=82x = 8 and x=4x = 4. The one-sided version gives x=5.5x = 5.5, and 2(5.5)+3=14≠112(5.5) + 3 = 14 \neq 11.

  • Trap Subtracting the coefficient instead of dividing, turning 6x=426x = 42 into x=36x = 36.

    Fix 6x6x means 6⋅x6 \cdot x, not 6+x6 + x, so divide both sides by 66: x=7x = 7.

  • Trap Dividing −3x=−6-3x = -6 by a bare 33 and writing x=−2x = -2.

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

  • Trap Letting a leading minus reach only the first term, so 9−(x+4)9 - (x + 4) becomes 9−x+49 - x + 4.

    Fix The minus is a factor of −1-1 on every term inside: 9−x−4=5−x9 - x - 4 = 5 - x.

  • Trap Adding the exponents or the letters when combining, writing 2x+3x=5x22x + 3x = 5x^2 or 3x+4y=7xy3x + 4y = 7xy.

    Fix Add coefficients only and keep the variable part: 2x+3x=5x2x + 3x = 5x. Unlike terms do not have identical variable parts, so their coefficients cannot be added; 3x+4y3x + 4y is already simplest.

  • Trap Reading "five less than xx" left to right as 5−x5 - x.

    Fix It is x−5x - 5: start at xx and take five away. At x=2x = 2 the two differ, −3-3 against 33.

  • Trap Dropping the parentheses around a negative input, so −x2-x^2 at x=−6x = -6 comes out as 3636.

    Fix −x2-x^2 squares the value first and negates after: −(−6)2=−36-(-6)^2 = -36. Only (−6)2(-6)^2, with the sign inside, is 3636.

Chapter Test Questions from across the chapter