Inequalities, Systems of Equations, and Decision-Making

An inequality compares quantities that need not be equal. The symbols < and > mean strictly less than and strictly greater than. The symbols ≤ and ≥ include equality. Translate 'at most' as ≤ and 'at least' as ≥.

Solve much as you solve an equation: add or subtract the same quantity on both sides. Multiplying or dividing both sides by a positive number preserves the inequality direction. Multiplying or dividing by a negative number reverses it because the order of numbers reverses.

A solution set contains every value making the original statement true. On a number line, use an open endpoint for < or > and a closed endpoint for ≤ or ≥. Shade toward smaller or larger values as appropriate.

Context can narrow the algebraic solution set. A budget may give q ≤ 13.75, but participant counts must be nonnegative whole numbers. The largest feasible count is therefore 13, and checking the next integer confirms why rounding to 14 would violate the limit.

A system asks for values satisfying two or more equations at once. With two variables, a solution is an ordered pair. Graphically, it lies on both lines, so an intersection represents a shared solution.

Substitution replaces a variable with an equal expression from another equation. When both plans are written as a total cost, setting their cost expressions equal is substitution: the same output must equal both expressions.

Elimination adds or subtracts equations to remove a variable. First multiply an equation if necessary to create equal or opposite coefficients. After finding one variable, substitute back to find the other, and check both original equations.

Two distinct nonparallel lines meet once. Parallel distinct lines have no common solution, while two equations describing the same line have infinitely many solutions. An algebraic result such as 0 = 5 signals inconsistency; a true identity such as 0 = 0 signals dependence when the original equations describe the same line.

An intersection is not automatically the best choice. Compare prices before and after it and consider restrictions such as capacity, included features, and allowed inputs. A mathematical solution outside the permitted domain may not be usable.

A practical plan may have several limits at once. Here x + y ≤ 8 limits total packs, while 2x + y ≤ 12 limits machine time. Also x ≥ 0 and y ≥ 0, and pack counts must be whole numbers.

To graph an inequality in two variables, first draw its boundary line. Use a solid line when equality is allowed and a dashed line for a strict inequality. Test a point not on the line to determine which side to shade. The overlap of all permitted regions is the feasible region.

The objective is the quantity you want to maximize or minimize. Here contribution is Z = 15x + 10y. Contribution is not necessarily final profit: any fixed expenses not included in these amounts would still need to be subtracted.

For a linear objective over a nonempty bounded polygonal feasible region, a maximum occurs at a corner; sometimes an entire edge ties. Find corners by intersecting boundary lines and axes, discard infeasible intersections, and evaluate the objective at the remaining corners.

Whole-number restrictions require care. A fractional best corner may be unusable, and rounding it can violate constraints or miss the best integer plan. In this example the best continuous corner already has whole-number coordinates, so it also solves the whole-pack problem.