CLEP College Mathematics Quick Review

Isolate the variable using inverse operations; inequalities behave like equations except when multiplying or dividing by a negative.

To solve $ax+b=c$: $x=/c-ba$. For inequalities, apply the same steps, but multiplying or dividing both sides by a negative number reverses the inequality symbol (e.g., $-2x<6 ⇒ x>-3$).

Don't confuse "solving an equation" (one exact numeric solution, uses $=$) with "solving an inequality" (a whole range of solutions, uses $<,>,≤,≥$).

Forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number.

$ax+b=c arrow x=/c-ba$; negative multiply/divide $arrow$ flip the sign.

A system's solution is where all equations are simultaneously true --- graphically, where the lines intersect.

Solve by substitution (solve one equation for a variable, plug into the other) or elimination (add/subtract to cancel a variable). A system has one solution, no solution (parallel lines), or infinitely many solutions (the same line).

Don't confuse "no solution" (same slope, different $y$-intercept) with "infinitely many solutions" (same slope, same $y$-intercept, the same line written twice).

Multiplying only one side of an equation by a constant when setting up elimination, instead of multiplying every term on both sides.

Same slope, different intercept $arrow$ no solution; same slope, same intercept $arrow$ infinite solutions.

A function assigns exactly one output to each input; $f(x)$ means ``plug $x$ into the rule $f$.''

To evaluate $f(a)$, substitute $a$ for every $x$ in the formula. Vertical line test: if a vertical line crosses the graph more than once, it is not a function.

Don't confuse $f(x)$ (function notation, read ``$f$ of $x$'') with $f· x$ (multiplication) --- the parentheses do not imply multiplication.

Misapplying order of operations after substitution, especially when squaring a negative input, e.g., $f(-2)$ for $f(x)=x^2$.

$f(a)$ = plug $a$ into the rule for every $x$.

Adding/subtracting inside or outside a function's parentheses shifts its graph horizontally or vertically.

$f(x)+k$ shifts up $k$; $f(x)-k$ shifts down $k$; $f(x+h)$ shifts left $h$; $f(x-h)$ shifts right $h$; $-f(x)$ reflects over the $x$-axis.

Don't confuse shifts inside the parentheses (horizontal, direction is opposite the sign) with shifts outside (vertical, direction matches the sign).

Assuming $f(x+2)$ shifts the graph right; it actually shifts the graph left.

Inside parentheses $arrow$ horizontal & opposite direction; outside $arrow$ vertical & same direction.

A linear function changes at a constant rate (its slope) and graphs as a straight line.

Slope $m=/y_2-y_1x_2-x_1$. Slope-intercept form: $y=mx+b$. Parallel lines share the same slope; perpendicular lines have slopes that are negative reciprocals ($m_1 m_2=-1$).

Don't confuse slope-intercept form $y=mx+b$ with point-slope form $y-y_1=m(x-x_1)$ --- both describe the same line from different known information.

Swapping the $x$- and $y$-coordinates when computing slope, which inverts or changes the sign of the true value.

$m=/riserun$; parallel $arrow$ same $m$; perpendicular $arrow$ negative reciprocal $m$.

Exponential models change by a constant percentage each period rather than a constant amount.

Growth: $y=a(1+r)^t$; decay: $y=a(1-r)^t$, where $a$ is the initial amount, $r$ is the rate as a decimal, and $t$ is time.

Don't confuse linear growth (adds a fixed amount each period, $y=a+bt$) with exponential growth (multiplies by a fixed factor each period, $y=a(1+r)^t$).

Plugging a percentage in as a whole number (e.g., 5) instead of converting to a decimal (0.05) before applying $(1± r)^t$.

Growth $arrow (1+r)^t$; decay $arrow (1-r)^t$.

Any quadratic $ax^2+bx+c=0$ can be solved with the quadratic formula, even when it does not factor nicely.

$x=/-b±sqrtb^2-4ac2a$. The discriminant $b^2-4ac$ predicts the number of real solutions: positive $arrow$ two, zero $arrow$ one, negative $arrow$ none real.

Don't confuse the discriminant $b^2-4ac$ (predicts the number/type of solutions) with the quadratic formula itself (finds the actual solution values).

Dropping the $±$ sign, which reports only one of the two possible solutions.

$x=/-b±sqrtb^2-4ac2a$; discriminant sign $arrow$ number of real roots.

Percent problems all reduce to ``part = percent $×$ whole,'' including increases and decreases.

Percent change $=/New-OldOld×100%$. Markup: New price $=$ Old price$×(1+r)$. Discount: New price $=$ Old price$×(1-r)$.

Don't confuse percent change (relative to the original value) with a percentage-point difference (a plain subtraction of two percentages).

Dividing by the new value instead of the original value when computing percent change.

%change $=/New-OldOld×100%$.

Simple interest grows by the same dollar amount every period because only the original principal earns interest.

$I=Prt$, where $P$ is principal, $r$ is the annual rate (decimal), $t$ is time in years. Total amount: $A=P(1+rt)$.

Don't confuse simple interest (only the original principal earns interest) with compound interest (interest also earns interest).

Using a rate given as a whole-number percent (e.g., 5) directly in the formula instead of converting to a decimal (0.05).

$I=Prt$; $A=P(1+rt)$.

Compound interest reinvests earned interest, so balances grow faster than under simple interest.

$A=P≤ft(1+/rn)^nt$, where $n$ is the number of compounding periods per year and $t$ is time in years.

Don't confuse the stated annual rate $r$ with the periodic rate $/rn$ actually used inside the formula.

Using $t$ alone as the exponent instead of $nt$ when compounding happens more than once per year.

$A=P(1+r/n)^nt$; more compounding periods $arrow$ more growth.

As compounding frequency grows without bound, the growth formula approaches $A=Pe^rt$.

Continuous compounding: $A=Pe^rt$, where $e≈2.71828$.

Don't confuse continuous compounding ($A=Pe^rt$) with ordinary compound interest ($A=P(1+r/n)^nt$) --- continuous is the limiting case as $n→∞$.

Plugging an $n$ value into a continuous-compounding problem instead of switching to $e^rt$.

Continuous $arrow A=Pe^rt$.

The effective annual rate reveals the true yearly growth rate once compounding is accounted for, and is always $≥$ the stated nominal rate.

$EAR=≤ft(1+/rn)^n-1$, where $r$ is the nominal annual rate (APR) and $n$ is the number of compounding periods per year.

Don't confuse APR (the stated nominal annual rate) with EAR/APY (the actual rate earned or paid after compounding).

Reporting the nominal APR as the final answer when the question asks for the effective (true) annual rate.

$EAR=(1+r/n)^n-1 ≥ APR$.

Present value and future value move money across time using the same compound-growth relationship, solved in opposite directions.

Future value: $FV=PV(1+r)^t$. Present value: $PV=/FV(1+r)^t$.

Don't confuse present value (what a future amount is worth today) with future value (what today's amount will grow to).

Multiplying by $(1+r)^t$ instead of dividing when the problem asks for present value.

$PV=/FV(1+r)^t$; $FV=PV(1+r)^t$.

Profit is revenue minus cost, and break-even is the point where revenue exactly equals cost.

Profit $=$ Revenue $-$ Cost. Break-even: set Revenue $=$ Cost and solve for the number of units.

Don't confuse markup based on cost with margin based on selling price --- they use different denominators and give different percentages for the same dollar amount.

Computing a markup percentage on cost when the question asks for a margin percentage based on selling price (or vice versa).

Profit $=$ Revenue $-$ Cost; break-even $arrow$ Revenue $=$ Cost.

Every data-display question is really asking you to read values or compare categories correctly off the visual.

Bar graphs compare categories; line graphs show change over time; circle (pie) graphs show parts of a whole (percentages sum to 100%); scatterplots show the relationship between two numeric variables; histograms show frequency of numeric data grouped into intervals.

Don't confuse a histogram (numeric data grouped into intervals, bars touch) with a bar graph (categorical data, bars usually separated).

Misreading the axis scale/interval size, causing an off-by-one-bar or off-by-one-interval comparison error.

Pie $arrow$ parts of a whole; scatterplot $arrow$ relationship; histogram $arrow$ frequency by interval.

Mean, median, and mode each describe a data set's ``center'' differently, and outliers affect them differently.

Mean $=/ xn$. Median $=$ the middle value when data are ordered (average the two middle values if $n$ is even). Mode $=$ the most frequent value.

Don't confuse the mean (pulled toward outliers/skew) with the median (resistant to outliers).

Finding the median without first sorting the data, or forgetting to average the two middle values for an even-sized data set.

Mean $=/ xn$; median = middle value (sorted); mode = most frequent.

Range and standard deviation both measure spread, but standard deviation uses every data point, not just the extremes.

Range $=$ Max $-$ Min. Standard deviation $=sqrt/ (x-)^2n$ measures the typical distance of data points from the mean.

Don't confuse range (uses only the two extreme values) with standard deviation (uses every value's distance from the mean).

Stopping after averaging the squared deviations (variance) and forgetting to take the final square root to get $$.

Range $=$ Max $-$ Min; $$ = typical distance from the mean.

In a normal distribution, data cluster symmetrically around the mean, and the empirical rule describes how much data falls within each standard deviation.

Empirical (68--95--99.7) rule: about 68% of data fall within $1$ of the mean, 95% within $2$, and 99.7% within $3$. The curve is symmetric with mean $=$ median $=$ mode.

Don't confuse a normal (symmetric, bell-shaped) distribution with a skewed distribution, where the mean and median differ and the mean is pulled toward the tail.

Assuming any bell-shaped-looking data set is exactly normal without checking that it is actually symmetric.

68--95--99.7 within 1, 2, 3 standard deviations of the mean.

A scatterplot's overall pattern shows whether --- and how strongly --- two variables move together.

Positive correlation: as $x$ increases, $y$ tends to increase (trend up-right). Negative correlation: as $x$ increases, $y$ tends to decrease (trend down-right). No clear pattern $arrow$ little or no correlation.

Don't confuse correlation (a statistical association between two variables) with causation (one variable directly causing the other) --- a strong correlation never proves causation.

Concluding a cause-and-effect relationship exists just because a scatterplot shows a strong correlation.

Up-right trend $arrow$ positive; down-right $arrow$ negative; correlation $$ causation.

Every logical statement is either true or false, and its negation always has the opposite truth value.

The negation of ``all $A$ are $B$'' is ``some $A$ are not $B$'' (not ``no $A$ are $B$''). The negation of ``some $A$ are $B$'' is ``no $A$ are $B$.''

Don't confuse the negation of ``all'' (``some… not'') with the opposite extreme (``none'').

Negating ``all students passed'' as ``no students passed'' instead of the correct ``at least one student did not pass.''

Negate ``all'' $arrow$ ``some… not''; negate ``some'' $arrow$ ``none.''

A conditional statement and its contrapositive always share the same truth value; the converse and inverse do not.

Conditional: $parrow q$. Converse: $qarrow p$. Inverse: $ parrow q$. Contrapositive: $ qarrow p$ (logically equivalent to the original conditional).

Don't confuse the converse (swap only) with the contrapositive (swap AND negate) --- only the contrapositive is guaranteed logically equivalent to the original.

Assuming a true conditional guarantees its converse is also true (the classic converse error).

$parrow q qarrow p$ (contrapositive only).

``And'' statements are true only when both parts are true; ``or'' statements are true when at least one part is true.

Conjunction $p q$ is true only if both $p$ and $q$ are true. Disjunction $p q$ is true if at least one of $p,q$ is true (false only when both are false).

Don't confuse the inclusive ``or'' of logic/math ($p q$, true if either or both) with the everyday exclusive ``or'' (one or the other, not both).

Marking $p q$ false when both $p$ and $q$ are actually true --- $p q$ is true whenever at least one is true.

$p q$: both true; $p q$: at least one true.

A Venn diagram is a picture of how sets overlap, and every region corresponds to a specific combination of membership.

$A B$ means every element of $A$ is also in $B$. The empty set $$ is a subset of every set. A set with $n$ elements has $2^n$ subsets.

Don't confuse $$ (element of) with $$ (subset of): $2\1,2,3\$ is true, but the correct subset statement is $\2\\1,2,3\$, not $2\1,2,3\$.

Treating $$ and $$ as interchangeable when labeling the relationship between an element and a set.

$$ = element in a set; $$ = one set inside another; $n$ elements $arrow 2^n$ subsets.

Union combines, intersection overlaps, and complement is ``everything else'' relative to the universal set.

$A B$ = elements in $A$ or $B$ (or both). $A B$ = elements in both $A$ and $B$. $A'$ (complement) = elements of the universal set $U$ not in $A$. $|A B|=|A|+|B|-|A B|$.

Don't confuse union $$ (combine everything from both sets) with intersection $$ (only the shared overlap).

Double-counting the overlap by adding $|A|+|B|$ without subtracting $|A B|$ to find $|A B|$.

$|A B|=|A|+|B|-|A B|$.

If a task is made of independent stages, multiply the number of choices at each stage to get the total number of outcomes.

If stage 1 has $m$ choices and stage 2 has $n$ choices, there are $m× n$ total outcomes; extend by multiplying across all stages of the task.

Don't confuse the multiplication principle (sequential/independent stages of one task, multiply) with the addition principle (mutually exclusive alternative choices, add).

Adding the number of choices at each stage instead of multiplying them together.

Sequential independent choices $arrow$ multiply.

A permutation counts arrangements where order matters; a combination counts selections where order does not.

Permutations: $_nP_r=/n!(n-r)!$. Combinations: $_nC_r=nr=/n!r!(n-r)!$.

Don't confuse a permutation (order matters --- ranking, arranging, distinct roles) with a combination (order doesn't matter --- selecting a group); $_nC_r$ is always $≤\ _nP_r$ for the same $n,r$.

Using the combination formula on a problem that specifies distinct positions or order (e.g., 1st, 2nd, 3rd place), overcounting or undercounting the result.

Order matters $arrow$ permutation; order doesn't $arrow$ combination.

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain).

$P(E)=/favorable outcomestotal outcomes$. The probability of the complement: $P(not E)=1-P(E)$.

Don't confuse the probability of an event $P(E)$ with the number of favorable outcomes alone --- probability is always a ratio (a fraction of the total), not a raw count.

Forgetting to reduce the sample space when outcomes are removed (e.g., drawing without replacement) and using the original total instead of the adjusted total.

$P(E)=/favorabletotal$; $P(not E)=1-P(E)$.

``Independent'' vs. ``conditional'' changes how you combine two events' probabilities, and expected value averages outcomes weighted by their likelihood.

Independent events: $P(A and B)=P(A)· P(B)$. Conditional probability: $P(A B)=/P(A and B)P(B)$. Expected value: $E(X)= x· P(x)$.

Don't confuse independent events (one event's occurrence doesn't affect the other, multiply straight through) with dependent/conditional events (the probability of one changes given the other occurred).

Multiplying $P(A)· P(B)$ for events that are actually dependent, such as drawing two cards without replacement.

Independent $arrow P(A)P(B)$; conditional $arrow /P(A and B)P(B)$.

A right triangle's side lengths are locked together by the Pythagorean theorem, and every triangle's area depends on base and height.

Area $=/12 bh$. Pythagorean theorem (right triangles only): $a^2+b^2=c^2$, where $c$ is the hypotenuse. Interior angles of any triangle sum to $180^$.

Don't confuse the hypotenuse (the side opposite the right angle, always the longest side) with the two legs in the Pythagorean theorem.

Applying $a^2+b^2=c^2$ to a triangle that is not confirmed to be a right triangle.

Right triangle $arrow a^2+b^2=c^2$; any triangle area $=/12 bh$.

Every quadrilateral area formula comes from decomposing the shape into rectangles or triangles.

Rectangle: $A=lw$, $P=2(l+w)$. Parallelogram: $A=bh$. Trapezoid: $A=/12(b_1+b_2)h$.

Don't confuse a parallelogram's height (the perpendicular distance between the parallel sides) with the length of its slanted side.

Using the slanted side length in place of the true perpendicular height in $A=bh$.

Rectangle $A=lw$; parallelogram $A=bh$; trapezoid $A=/12(b_1+b_2)h$.

A circle's circumference and area both scale from its radius through $$, but circumference is linear in $r$ while area is quadratic.

Circumference $C=2 r= d$. Area $A= r^2$.

Don't confuse circumference (a perimeter/length, uses $2 r$) with area (a two-dimensional measure, uses $ r^2$).

Plugging the diameter directly into the area formula ($ d^2$) instead of first halving it to get the radius.

$C=2 r$; $A= r^2$.

When a transversal crosses parallel lines, it creates predictable equal or supplementary angle pairs.

Corresponding angles and alternate interior angles are equal when lines are parallel. Perpendicular lines meet at $90^$ and have slopes that are negative reciprocals. Supplementary angles sum to $180^$; complementary angles sum to $90^$.

Don't confuse complementary angles (sum to $90^$) with supplementary angles (sum to $180^$).

Assuming same-side (co-interior) angles are equal --- they are actually supplementary, not congruent.

Parallel + transversal $arrow$ corresponding/alternate angles equal; complementary $=90^$, supplementary $=180^$.

Every real number belongs to a nested hierarchy of number sets, from natural numbers up through the reals.

$NWZQR$ (naturals $$ whole numbers $$ integers $$ rationals $$ reals); irrational numbers (e.g., $sqrt2,$) are real numbers that are not rational.

Don't confuse rational numbers (can be written as $/ab$ with integers $a,b$; terminating or repeating decimals) with irrational numbers (non-terminating, non-repeating decimals).

Classifying a repeating decimal like $0.3$ as irrational when it is actually rational (it equals $/13$).

$NWZQR$; irrationals live in $R$ only.

Prime factorization is the building block for finding greatest common factors and least common multiples.

A prime number has exactly two factors (1 and itself); 2 is the only even prime. GCF = product of shared prime factors at their lowest shared powers; LCM = product of all prime factors involved at their highest powers.

Don't confuse the GCF (Greatest Common Factor, divides into both numbers, always $≤$ the smaller number) with the LCM (Least Common Multiple, both numbers divide into it, always $≥$ the larger number).

Treating 1 as a prime number, or forgetting that 2 is prime even though it is even.

GCF $≤$ smaller number; LCM $≥$ larger number.

Absolute value measures distance from zero on the number line, so it is always non-negative.

$|x| = x$ if $x≥0$; $|x|=-x$ if $x<0$. For $a>0$, the equation $|x|=a$ has two solutions: $x=a$ or $x=-a$.

Don't confuse $|x|=a$ for $a>0$ (two solutions, $± a$) with $|x|=-a$ for $a>0$ (no solution, since absolute value can never be negative).

Reporting only one of the two solutions ($x=a$) when solving an absolute value equation, and dropping the negative case.

$|x|=a arrow x=± a$ (for $a>0$).

Converting units is just multiplying by well-chosen fractions equal to 1 (conversion factors) so the old unit cancels.

Multiply by a conversion factor $/new unitold unit=1$, e.g., $5 ft×/12 in1 ft=60 in$.

Don't confuse converting a length (one dimension, one conversion factor) with converting an area or volume (two or three dimensions, so the conversion factor must be squared or cubed).

Forgetting to square (or cube) the conversion factor when converting square units (area) or cubic units (volume).

Multiply by conversion fraction $=1$ so units cancel; area/volume $arrow$ square/cube the factor.

This box is your one-page emergency formula sheet.

When two terms sound alike, ask: does order/spread/direction matter?

$NWZQR$; $$ combines, $$ overlaps, complement excludes.

Know the symbol before test day --- there's no time to decode it during the exam.

Classify first, formula second, units always --- then solve.

60 questions, 90 minutes, all MC, on-screen TI-30XS MultiView, no guessing penalty.

Mean $arrow$ deviations $arrow$ square $arrow$ sum $arrow$ divide by $n$ $arrow$ square root.

Circle the compounding keyword first --- it tells you which formula to use.

If a rule ``sounds obvious,'' check it against a quick numeric example before trusting it.