Free CLEP College Math Exam Flashcards

Reverse the inequality sign. Dividing -3x > 12 by -3 gives x < -4, not x > -4. Adding or subtracting a negative never flips the sign; only multiplying or dividing by one does.

x = 4 and y = 3. Adding the two equations eliminates y because the coefficients are opposites: 3x = 12, so x = 4. Substituting back gives 4 - y = 1, so y = 3.

f(-2) = (-2)^2 - 4(-2) = 4 + 8 = 12. The common error is treating (-2)^2 as -4. Substitute the whole value, parentheses included, before applying the exponent.

y = f(x - 3) shifts the graph 3 units right; y = f(x) + 3 shifts it 3 units up. A change inside the function moves the graph horizontally and in the direction opposite the sign.

y = -f(x) reflects the graph across the x-axis, flipping it top to bottom. y = f(-x) reflects it across the y-axis, flipping it left to right. The negative outside acts on outputs, inside on inputs.

If f(-x) = f(x) the function is even and symmetric about the y-axis, like y = x^2. If f(-x) = -f(x) it is odd and symmetric about the origin, like y = x^3. Most functions are neither.

Linear growth adds a constant amount each step, as in 3, 6, 9, 12. Exponential growth multiplies by a constant ratio, as in 3, 6, 12, 24. Check the differences first; if they vary, check the ratios.

Slope = (11 - 3) / (6 - 2) = 8 / 4 = 2. Slope is change in y over change in x. Subtract the points in the same order top and bottom, or the sign comes out wrong.

For ax^2 + bx + c = 0 with a not equal to 0, x = (-b +/- sqrt(b^2 - 4ac)) / (2a). The discriminant b^2 - 4ac sets the count: positive gives two real roots, zero gives one, negative gives no real roots.

Different slopes give exactly one solution where the lines cross. Equal slopes with different y-intercepts give none, because parallel lines never meet. Identical equations give infinitely many, because the lines coincide.

I = Prt, so I = 3000 x 0.06 x 2 = $360. Simple interest is always computed on the original principal, so it never earns interest on interest already credited.

A = P(1 + r/n)^(nt), where n is compounding periods per year and t is years. For $1,000 at 8% compounded quarterly for 1 year: 1000(1.02)^4 = $1,082.43. A larger n produces a larger balance.

A = Pe^(rt), where e is about 2.718. Continuous compounding is the limit as the number of periods grows without bound, so at the same stated rate it always beats annual, monthly, or daily compounding, though its edge over daily is tiny.

Effective annual yield = (1 + r/n)^n - 1. A 6% nominal rate compounded monthly gives (1.005)^12 - 1, or about 6.17%. The effective yield exceeds the stated rate whenever interest compounds more than once a year.

Percent change = (new value - old value) / old value, times 100. The frequent error is dividing by the new value. The base is always the original amount, for an increase or a decrease alike.

40%, not 45%. Successive discounts multiply: 0.75 x 0.80 = 0.60 of the original price, so 40% comes off. The second discount applies to the already-reduced price, so percents never simply add.

Markup on cost is 15/60 = 25%. Profit margin on selling price is 15/75 = 20%. Both use the same $15 profit but different bases, so always check whether a percent is quoted on cost or on price.

PV = FV / (1 + r)^t. To have $1,102.50 in 2 years at 5%, you need 1102.50 / 1.05^2 = $1,000 today. Present value discounts backward in time; future value compounds forward.

Simple interest earns $200, for a $1,200 balance. Compounded annually it earns $210, for $1,210. The $10 gap is the second year's interest earned on the first year's interest.

No. 1.10 x 0.90 = 0.99, so the price ends 1% below where it started. Equal percent increases and decreases do not cancel, because the decrease is taken on the larger, raised amount.

The median. For 2, 3, 4, 5, 100 the mean is 22.8 but the median is 4. One extreme value pulls the mean toward it, while the median depends only on position in the sorted list.

The mode is the most frequently occurring value. A set can have no mode when every value appears once, exactly one mode, or several when values tie for highest frequency. It is the only center usable for non-numeric data.

Range = largest value - smallest value, reported as one number rather than the pair of endpoints. It measures spread but uses only the two extremes, so a single outlier can inflate it dramatically.

The mean rises by 5 and the standard deviation does not change, because every value moves the same distance and spread depends only on the gaps between values. Multiplying every value by 2 instead doubles both the mean and the standard deviation.

Standard deviation measures the typical distance of values from the mean, and it equals zero only when every value is identical. CLEP College Mathematics asks conceptual questions about it rather than requiring you to compute the formula.

About 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. The rule applies only to roughly symmetric bell-shaped data, not to skewed data.

A circle or pie graph shows parts of a whole, and its slices must total 100%. A scatterplot shows the relationship between two numeric variables. A histogram shows how one variable is distributed across bins.

Mean = 66 / 6 = 11. Median = 9: with an even count, average the two middle values of the sorted list, (8 + 10) / 2. Mode = 8, the only repeated value. Always sort before locating the middle, or the median comes out wrong.

Only the contrapositive, 'If not q, then not p'. The converse 'If q, then p' and the inverse 'If not p, then not q' are not equivalent to the original, though they are equivalent to each other.

'Some swan is not white', meaning at least one exists. It is not 'No swans are white'. Because of this, a single counterexample is enough to disprove any universal statement.

A conjunction 'p and q' is true only when both parts are true. An inclusive disjunction 'p or q' is false only when both parts are false, so it stays true when both parts happen to hold.

Only when p is true and q is false. If the hypothesis p is false, the conditional counts as true no matter what q is, which is called vacuously true. A false premise never makes the statement false.

A union B contains every element in A, in B, or in both. A intersection B contains only elements in both. The complement of A contains every element of the universal set not in A. Disjoint sets have an empty intersection.

2^4 = 16. The count is 2^n because each element is independently either in or out. The empty set and the set itself both count as subsets, so there are 15 proper subsets.

4 students. Use n(A or B) = n(A) + n(B) - n(A and B) = 18 + 15 - 7 = 26 taking at least one, so 30 - 26 = 4 take neither. Subtract the overlap once or you double-count it.

If one choice has m options and an independent second choice has n options, the pair has m x n outcomes. With 4 shirts, 3 pants, and 2 pairs of shoes there are 4 x 3 x 2 = 24 outfits.

Use a combination when order does not matter and a permutation when it does. C(n,r) = n! / (r!(n-r)!) and P(n,r) = n! / (n-r)!. From 6 people, C(6,2) = 15 committees but P(6,2) = 30 ordered finishes.

P(A or B) = P(A) + P(B) - P(A and B). Subtract the overlap so it is not counted twice. For mutually exclusive events that overlap is 0, and complementary events satisfy P(A) + P(not A) = 1.

Without replacement the draws are dependent. From 3 red and 5 blue marbles, P(two reds) = (3/8)(2/7) = 3/28, because the second draw has one fewer red and one fewer marble. With replacement both factors stay 3/8.

Expected payout = 0.2(10) + 0.8(0) = $2, so the net expected value is 2 - 3 = -$1 per ticket. Expected value is a long-run average per trial, not an outcome any single play can produce.

a^2 + b^2 = c^2, valid only in a right triangle, where c is the hypotenuse opposite the right angle. Legs of 5 and 12 give a hypotenuse of 13, since 25 + 144 = 169.

Corresponding, alternate interior, and alternate exterior angles are equal. Same-side (consecutive) interior angles are supplementary, summing to 180 degrees. Perpendicular lines meet at 90 degrees, and two lines each perpendicular to the same line are parallel to each other.

Circumference = 2(pi)r and area = (pi)r^2. A sector with central angle t degrees has area (t/360)(pi)r^2. The usual error is substituting the diameter for r, so halve the diameter first.

The inscribed angle is half the central angle that subtends the same arc. As a consequence, any angle inscribed in a semicircle measures 90 degrees, because the diameter's central angle is 180 degrees.

1 to 9. Similar figures have equal corresponding angles and sides scaled by a factor k, so perimeters scale by k but areas scale by k^2. Here k = 3, so the area ratio is 3^2 = 9.

A rational number can be written as a ratio of two integers, so its decimal terminates or repeats. sqrt(9) = 3 is rational. sqrt(2) and pi are irrational because their decimals never terminate or repeat.

Every integer greater than 1 factors into primes in exactly one way apart from order, as in 60 = 2^2 x 3 x 5. The number 1 is neither prime nor composite, and 2 is the only even prime.

4.7 x 10^-4. Proper scientific notation requires the coefficient a to satisfy 1 <= |a| < 10. Moving the decimal to the right makes the exponent negative, and 47 x 10^-5 is not in proper form.

Absolute value is distance from zero, so it is never negative. |x| = 7 gives x = 7 or x = -7. The expression |x - 3| is the distance between x and 3, and |x| = -2 has no solution.

GCF = 6 and LCM = 36. From 12 = 2^2 x 3 and 18 = 2 x 3^2, the GCF takes the lowest power of each shared prime and the LCM takes the highest power of every prime that appears.