Papa's Freezeria — CalcBot

On the number line, the number to the right is always greater. Positive numbers are greater than zero, and zero is greater than negative numbers. For any two real numbers a and b:

if a − b > 0, then a > b;   if a − b < 0, then a < b;   if a − b = 0, then a = b.

When comparing negative numbers, the one with the smaller absolute value is greater: −3 > −7 because |−3| < |−7|. To compare fractions, convert them to a common denominator or to decimal form. For example, to compare 3/7 and 2/5: 3/7 = 15/35 and 2/5 = 14/35, so 3/7 > 2/5.

Key rules: (1) If a > b and b > c, then a > c (transitivity). (2) If a > b, then a + c > b + c for any c. (3) If a > b and c > 0, then ac > bc; but if c < 0, then ac < bc — multiplying by a negative reverses the inequality. (4) The reciprocal of a larger positive number is smaller: if a > b > 0, then 1/a < 1/b.

A common fraction a/b (where b ≠ 0) represents parts of a whole. Equivalent fractions represent the same value: a/b = (a·k)/(b·k) for k ≠ 0. Reducing a fraction means dividing numerator and denominator by their greatest common divisor (GCD). For example, 18/24 = (18÷6)/(24÷6) = 3/4.

Decimal fractions use place values: 0.75 = 75/100 = 3/4. To compare decimal fractions, align decimal points and compare digit by digit. A terminating decimal is a fraction with denominator a power of 10; a repeating decimal like 0.3̄ = 3/9 = 1/3. The general rule: 0.ᾱ = a/9, 0.āb̄ = (10a + b)/99.

To add or subtract fractions, find the least common denominator (LCD) — the LCM of the denominators. For example: 1/6 + 3/8: LCM(6,8) = 24, so 1/6 + 3/8 = 4/24 + 9/24 = 13/24. To multiply: (a/b)·(c/d) = (ac)/(bd). To divide: (a/b) ÷ (c/d) = (ad)/(bc), where c ≠ 0.

Two quantities x and y are inversely proportional if their product is constant: x · y = k, or equivalently y = k/x, where k ≠ 0 is the constant of inverse proportionality. The graph of y = k/x is a hyperbola with two branches in quadrants I and III (when k > 0) or II and IV (when k < 0). As x increases, y decreases, and vice versa — but they never reach zero.

Key properties: (1) If x is multiplied by a factor n, then y is divided by n (e.g., if x doubles, y halves). (2) The product of any pair of corresponding values is always k. (3) The function y = k/x is undefined at x = 0 and never crosses either axis.

Typical problems: If 6 workers complete a job in 10 days, how many days for 15 workers? Since workers × days = constant: 6 × 10 = 15 × d, giving d = 4 days. Similarly, if a car traveling at 60 km/h takes 3 hours, at 90 km/h it takes 60×3/90 = 2 hours. Always verify that the product remains constant.

A linear inequality with a parameter has the form ax + b > 0 (or <, ≤, ≥). The solution depends on the parameter a: if a > 0, then x > −b/a; if a < 0, the inequality sign reverses: x < −b/a; if a = 0 and b > 0, every x is a solution; if a = 0 and b ≤ 0, there is no solution. Always analyze the critical parameter values where the coefficient of x changes sign.

For quadratic inequalities ax² + bx + c > 0, first find the discriminant D = b² − 4ac. If D < 0 and a > 0, the entire parabola is above the x-axis — all real x satisfy the inequality. If D > 0, find roots x₁ and x₂, then the sign of ax² + bx + c matches the sign of a outside the interval [x₁, x₂] and is opposite between the roots.

The graphical method is powerful: sketch the parabola y = ax² + bx + c and identify where it is above or below the x-axis. When parameters are involved, consider cases: D > 0 (two distinct roots), D = 0 (one double root), D < 0 (no real roots). For each case, determine the solution set based on the sign of a and the direction of the inequality.

A rational inequality involves a fraction with polynomials, such as P(x)/Q(x) > 0. The interval method (also called the sign chart method) is the standard approach: (1) Find all zeros of P(x) and Q(x). (2) Plot these points on a number line (use open circles for zeros of Q(x) since they are excluded from the domain). (3) Determine the sign of the expression in each interval. (4) Select intervals matching the inequality sign.

The rightmost interval always has the sign determined by the leading coefficients of P(x) and Q(x). Signs alternate when crossing a root of odd multiplicity but stay the same when crossing a root of even multiplicity. For example, for (x−1)²(x+2) / ((x−3)(x+1)) > 0, the sign does not change at x = 1 (even power) but does at x = −2, x = −1, and x = 3.

Common mistakes: (1) Multiplying both sides by a denominator without knowing its sign — this can reverse the inequality. (2) Forgetting that zeros of the denominator are never included in the solution. (3) Including roots of the numerator for strict inequalities (> or <) — these should be excluded. (4) Not checking whether the expression is defined at boundary points. Always bring everything to one side as a single fraction before applying the interval method.