Free ALEKS Math Exam Flashcards

Change subtraction of a negative into addition: a - (-b) = a + b. This sign rule is a common ALEKS arithmetic trap.

Evaluate grouping symbols first, then exponents, then multiplication/division left to right, then addition/subtraction left to right.

Use a common denominator before adding or subtracting fractions. Only the numerators combine after the denominators match.

Percent change = (new - original) / original x 100%, for a positive original value. Use the original value as the denominator.

A ratio comparison can be solved by cross-multiplying equivalent fractions, as long as the units line up consistently.

Absolute value is distance from zero. |x| = a gives two cases when a > 0: x = a or x = -a.

Undo operations in reverse order and perform the same operation on both sides. Check by substituting the result back in.

a(b + c) = ab + ac. Distribute the sign as well as the coefficient, especially when a is negative.

Only terms with the same variable part and exponent combine. 3x and 5x combine; 3x and 5x^2 do not.

When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality symbol.

For a nonvertical line, y = mx + b, where m is slope and (0, b) is the y-intercept.

For distinct x-coordinates, slope is (y2 - y1) / (x2 - x1): vertical change divided by horizontal change. Equal x-coordinates give a vertical line with undefined slope.

Nonvertical parallel lines have equal slopes. Perpendicular nonvertical lines have slopes that are negative reciprocals.

f(a) means substitute a for x in the rule f(x). It is an output value, not multiplication by f.

The domain is the allowed input set. For real-valued functions, exclude zero denominators, negative inputs to even roots, and nonpositive inputs to logarithms.

The range is the set of possible output values. Use the graph, vertex, or restrictions to identify what y-values occur.

An x-intercept occurs where y = 0. To find it algebraically, set the expression equal to zero and solve for x.

A y-intercept occurs where x = 0. Substitute x = 0 into the equation or read where the graph crosses the y-axis.

A quadratic function has form f(x) = ax^2 + bx + c, with a not equal to zero. Its parabola opens upward for a > 0 and downward for a < 0.

Before using other factoring methods, pull out the greatest common factor from every term.

a^2 - b^2 factors as (a - b)(a + b). It applies only to subtraction of two square terms.

When multiplying like bases, add exponents: x^a x^b = x^(a+b). The base must be the same.

For integer exponents where the expressions are defined, (x^a)^b = x^(ab). For arbitrary real exponents, a positive base avoids sign and domain exceptions.

A negative exponent means reciprocal: x^-a = 1/x^a for x not equal to zero.

Any nonzero base to the zero power equals 1. The expression 0^0 is not treated as a standard simplification.

For a nonzero polynomial in one variable, the degree is the greatest exponent with a nonzero coefficient after like terms are combined.

Values that make the denominator zero are excluded, even if a factor later cancels.

Factor numerator and denominator, cancel common factors, and keep the original denominator restrictions.

Find a common denominator, rewrite each fraction, combine numerators, then factor and simplify if possible.

Multiply by the least common denominator to clear fractions, then reject any solution that makes an original denominator zero.

The radical symbol gives the principal nonnegative square root. sqrt(25) = 5, not plus or minus 5.

For real-valued even roots, the radicand must be greater than or equal to zero.

x^(m/n) means the nth root of x raised to the m power, when the expression is defined in the real numbers.

Isolate the radical, raise both sides to the needed power, and check for extraneous solutions.

A common model is y = a(1 + r)^t, where r is the growth rate per period. Decay uses 1 - r.

For x > 0 and b > 0 with b not equal to 1, y = log_b(x) means b^y = x. The logarithm is the exponent.

log_b(xy) = log_b(x) + log_b(y), for positive x and y and base b > 0, b not equal to 1.

log_b(x/y) = log_b(x) - log_b(y), with positive arguments and a valid base.

For x > 0, b > 0, b not equal to 1, and real k, log_b(x^k) = k log_b(x). Check the argument and base before applying the rule.

Distance between (x1, y1) and (x2, y2) is sqrt((x2 - x1)^2 + (y2 - y1)^2).

The midpoint is ((x1 + x2)/2, (y1 + y2)/2). Average the x-values and y-values separately.

For a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse.

Area of a circle is pi r^2. Circumference is 2 pi r or pi d.

Area = 1/2(base)(height). The height must be perpendicular to the base.

A vertical line has undefined slope because the change in x is zero.

For an acute angle theta in a right triangle: sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, and tan(theta) = opposite/adjacent.

On the unit circle, a point at angle theta has coordinates (cos theta, sin theta).

180 degrees equals pi radians. Convert degrees to radians by multiplying by pi/180.

Solve one equation for a variable, substitute into the other equation, then back-substitute to find the remaining variable.

Practice producing the answer, not just recognizing it. The real ALEKS placement assessment uses open-response adaptive questions.