Free PERT Math Exam Flashcards

Addition reverses subtraction and multiplication reverses division. To isolate the variable, apply the opposite operation to both sides, preserving the equality.

Distribute first, combine like terms on each side, move variable terms to one side and constants to the other, then divide by the coefficient. Skipping the order leaves the variable hidden inside nested operations.

Collect the variable terms on one side by adding or subtracting, then collect the constants on the other. Choose the side that keeps the variable's coefficient positive to avoid sign errors.

Flip the inequality only when multiplying or dividing both sides by a negative number. Adding, subtracting, and multiplying or dividing by a positive number leave the direction unchanged.

An 'and' compound inequality such as -3 < x <= 5 requires both conditions to hold at once, so the solution is the overlap of the two number lines. Graph it as a connected segment between the two endpoints.

An 'or' compound inequality is satisfied when either condition holds, so the solution is the union of two rays. Graph it as two arrows pointing outward from the two boundary points.

Clear fractions first by multiplying every term by the LCD of all denominators. This converts a fraction-heavy equation into an integer-coefficient equation and avoids messy fraction arithmetic.

Substitute the candidate value back into the original equation; if both sides are equal, the value is a solution. With fractions or radicals, checking catches extraneous solutions introduced by squaring or multiplying by a variable expression.

Solve one equation for one variable, then substitute that expression into the other equation. This collapses two equations in two unknowns into one equation in one unknown.

Stack the equations aligned by variable, multiply one or both by constants so a variable's coefficients are opposites, then add the equations to drop that variable. Choose the variable whose coefficients are easiest to match.

A consistent system has at least one solution (one unique solution or infinitely many); an inconsistent system has none. Graphically, parallel distinct lines are inconsistent and overlapping lines give infinitely many solutions.

Only terms with the same variable raised to the same power can be combined by adding or subtracting their coefficients. 3x and 3x^2 are not like terms even though the variable letter matches.

a(b + c) = ab + ac; multiply the outside factor by each inside term. With a negative outside factor, the sign flips on each inside term: -(2x - 5) = -2x + 5.

First, Outer, Inner, Last: multiply two binomials term-by-term in this order, then combine like terms. For (x + 3)(x - 2), this gives x^2 - 2x + 3x - 6, which simplifies to x^2 + x - 6.

Parentheses, Exponents, Multiplication and Division left to right, Addition and Subtraction left to right. Multiplication and division share the same priority, as do addition and subtraction.

Find the largest factor common to every term, including the smallest power of each shared variable, and pull it outside a parenthesis. For 6x^3 + 9x^2, the GCF is 3x^2, giving 3x^2(2x + 3). Always factor the GCF first.

Factors as (a + b)(a - b), the sum and difference of the square roots. It only works when both terms are perfect squares and the operation between them is subtraction.

Factors as (a + b)^2. Verify by checking that the middle term is exactly twice the product of the square roots of the first and last terms; otherwise the trinomial is not a perfect square.

For ax^2 + bx + c, multiply a times c, find two numbers that multiply to ac and add to b, split the middle term using those numbers, then factor by grouping. The AC method rescues you when the leading coefficient is not 1.

Group terms in pairs, factor the GCF from each pair, then pull out the common binomial. It works whenever the two pairs share a binomial factor after the GCF is pulled from each.

If the product of factors is zero, at least one factor is zero: if (x - 2)(x + 5) = 0, then x = 2 or x = -5. This is why factoring solves quadratics; set each factor equal to zero.

Divide term-by-term from highest degree to lowest, subtract, and bring down the next term, exactly like numerical long division. If the remainder is zero, the divisor is a factor of the dividend.

ax^2 + bx + c = 0 with a not equal to zero. Setting the equation to zero is required before factoring or applying the quadratic formula; a non-zero right-hand side breaks both methods.

x = (-b plus or minus the square root of (b^2 - 4ac)) divided by (2a). It solves any quadratic, even ones that do not factor over the integers. The expression under the root is the discriminant.

Positive means two distinct real roots, zero means one repeated real root, and negative means two complex (non-real) roots. The sign tells you the shape of the x-intercepts before you solve.

When a quadratic has the form x^2 = k, take the square root of both sides: x = plus or minus the square root of k. Use it on quadratics with no linear term; it is faster than the full quadratic formula for that shape.

Move everything to one side so the expression equals zero, factor, then apply the zero product property to each factor. If you cannot find integer factors, switch to the quadratic formula.

y = a(x - h)^2 + k has its vertex at (h, k). The sign of a controls whether the parabola opens upward (a > 0) or downward (a < 0).

Factor numerator and denominator completely, then cancel factors that appear in both. You can only cancel common factors, never common terms; (x + 2)/x is not simplified by canceling the 2s.

Find the LCD of all denominators, rewrite each fraction with that LCD, then add numerators and keep the denominator. Multiplying denominators always gives a common denominator, but using the LCD keeps the arithmetic smaller.

Factor everything first, multiply numerators together and denominators together, then cancel common factors across the fraction bar. Always reduce before multiplying to avoid huge coefficients.

Keep the first fraction, change division to multiplication, and flip the second fraction (take its reciprocal). Then multiply and simplify as usual.

Any value that makes a denominator zero is excluded from the domain, even if it would cancel from the numerator. State these restrictions before simplifying, since canceling can hide them.

Multiply both sides by the LCD of all denominators to clear the fractions, then solve the resulting polynomial equation. Always check for extraneous solutions that make an original denominator zero.

a^m times a^n equals a^(m + n); add exponents when multiplying like bases. The bases must match; do not add exponents across different bases.

a^m divided by a^n equals a^(m - n); subtract exponents when dividing like bases. A negative result signals the answer belongs in the denominator as a positive exponent.

Multiply the exponents: (a^m)^n equals a^(mn). This differs from the product rule, which adds exponents of like bases being multiplied together.

a^(-n) equals 1 / a^n. A negative exponent signals a reciprocal, not a negative number; 2^(-3) equals 1/8, not -8.

a^(1/n) equals the n-th root of a; the denominator of the exponent becomes the index of the root. More generally, a^(m/n) equals the m-th power of the n-th root of a.

Factor out the largest perfect square from the radicand. The square root of 72 becomes the square root of (36 times 2), which equals 6 times the square root of 2. Stop only when the remaining radicand has no perfect-square factor.

Multiply numerator and denominator by the radical in the denominator to remove it from the bottom: 5 over the square root of 3 becomes (5 times the square root of 3) over 3. Use conjugate pairs when the denominator is a sum or difference of radicals.

y = mx + b, where m is the slope and b is the y-intercept. Reading the form directly tells you the line's steepness and where it crosses the y-axis.

m = (y2 - y1) / (x2 - x1). Keep the point order consistent in numerator and denominator; swapping the order of one point but not the other flips the sign of the slope.

Parallel lines have equal slopes. Perpendicular lines have negative reciprocal slopes: m1 times m2 equals -1, so a line with slope 2 is perpendicular to a line with slope -1/2.

d equals the square root of ((x2 - x1)^2 + (y2 - y1)^2). It is the Pythagorean theorem applied to the horizontal and vertical legs between two points on the plane.

Midpoint equals ((x1 + x2)/2, (y1 + y2)/2). It averages the x-coordinates and the y-coordinates separately to find the point exactly halfway between two endpoints.

For a right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2. It underpins the distance formula and only applies when the triangle has a 90-degree angle.

Percent change equals (new - old) divided by the absolute value of old, times 100. The denominator is always the original amount, not the new amount; confusing the two is the most common percent error.

Identify what x represents, write the relationship in English first, then translate keywords: 'of' means multiply, 'is' means equals, 'per' means divide, and 'less than' reverses the subtraction order.

When y varies directly with x, the ratio y/x is a constant k. Doubling x doubles y, tripling x triples y. The graph is a straight line through the origin with slope k.