Inequalities: Chapter Review

Vocabulary and notation

Inequality

A comparison of two quantities that need not be equal, using $<$, $>$, $\le$, or $\ge$ in place of an equals sign.

Strict and inclusive

$<$ and $>$ are strict and exclude the boundary value; $\le$ and $\ge$ are inclusive and admit it. That one bar is the whole difference.

Solution set

Every value making the inequality true. For a linear inequality it is almost always a whole range, infinitely many numbers, so it is drawn or written as an interval, not listed. The one exception is an empty solution set, where nothing works.

Half-plane

One of the two pieces a line cuts the plane into. The line itself is the border between them, called the boundary.

Constraint

One hard limit of an optimization problem, written as a linear inequality. $x \ge 0$ and $y \ge 0$ are constraints too, whenever the variables count real things.

Objective function

The single linear quantity $P = ax + by$ being pushed as high or as low as it will go, such as a profit to maximize or a cost to minimize.

Level line

The line $ax + by = c$ on which the objective holds the one value $c$. All level lines of an objective are parallel, so changing $c$ slides the line without turning it.

Feasible region

The overlap of every constraint's half-plane, the points satisfying all of them at once. It may be a bounded polygon (one fitting inside some box), an unbounded region running off forever, or empty.

Vertex (corner)

A point of the feasible region where two boundary lines cross.

Formulas and theorems

• The four symbols

  $<$ less than; $>$ greater than; $\le$ at most; $\ge$ at least.

  Use when The symbol opens toward the larger amount and points at the smaller. Only $\le$ and $\ge$ let the two sides be equal.

• Turning an inequality end for end

  $$a \le b \iff b \ge a$$

  Use when Swap the sides AND reverse the symbol; the meaning is untouched. All four symbols. This is how you move a variable to the left.

  e.g. $7 > x$ says the same thing as $x < 7$.

• Adding or subtracting keeps the direction

  $$a < b \implies a + c < b + c$$

  Use when Any real $c$. Addition and subtraction NEVER reverse the symbol, not even when $c$ is negative or the result is negative.

• Multiplying or dividing by a positive keeps the direction

  $$a < b \text{ and } c > 0 \implies ac < bc$$

  Use when $c > 0$ strictly. $c = 0$ is barred: it collapses both sides to $0$ and destroys the comparison.

• Multiplying or dividing by a negative reverses the direction

  $$a < b \text{ and } c < 0 \implies ac > bc$$

  Use when $c < 0$ strictly, and its sign must be KNOWN: multiplying by an expression of unknown sign, such as $x$ or $x - 2$, is not allowed at all, because neither direction can be justified. The reversal happens once, at that step only.

  e.g. $-\tfrac{1}{2}x < 4$ times $-2$ gives $x > -8$.

• Transitivity

  $$a < b \text{ and } b < c \implies a < c$$

  Use when Both comparisons must point the same way. It holds with $\le$ throughout, and mixing one $<$ with one $\le$ still gives the strict $a < c$.

  e.g. $2 < 5$ and $5 < 9$ give $2 < 9$.

• Endpoint conventions, in all three pictures

  Strict ($<$, $>$): open circle, parenthesis, dashed boundary. Inclusive ($\le$, $\ge$): filled circle, bracket, solid boundary.

  Use when Read it off the FINAL symbol after solving, not the one you started with. Infinity is not a number, so $\infty$ and $-\infty$ always take a parenthesis.

  e.g. $x \le 5$ is $(-\infty, 5]$; $x > 2$ is $(2, \infty)$.

• Which way a number line shades

  A "greater" relation ($>$, $\ge$) shades right, toward the larger numbers; a "less" relation ($<$, $\le$) shades left.

  Use when Get the variable alone on the left first, so rewrite $7 > x$ as $x < 7$ before drawing. Switching strict to inclusive never moves the shading, it only closes the circle.

  e.g. $x \ge -2$: filled circle on $-2$, shaded right, the interval $[-2, \infty)$.

• Compound inequalities: and versus or

  An "and" compound $a < x < b$ is the intersection, one band between the bounds. An "or" compound is the union of two separate solution sets, joined with $\cup$.

  Use when In $a < x < b$ both symbols must point the same way with the smaller bound on the left; $3 < x < -2$ describes nothing at all.

  e.g. $x < -1$ or $x \ge 2$ is $(-\infty, -1) \cup [2, \infty)$.

• When the variable cancels

  A true leftover means all real numbers, $(-\infty, \infty)$. A false leftover means no solution, an empty solution set.

  Use when Applies only when EVERY copy of the variable cancels, leaving no variable behind. Those two are the only outcomes, and each is a complete answer.

  e.g. $2x + 5 < 2x + 1$ collapses to $5 < 1$, which is false, so there is no solution.

• Boundary of a two-variable inequality

  Replace the symbol with $=$: that line is the boundary, and the solutions are exactly one of the two half-planes it separates, never a mix of both, together with the boundary line itself when the symbol is inclusive.

  Use when Any linear inequality in $x$ and $y$. A missing variable is no special case: $x > c$ gives a vertical boundary and $y \le c$ a horizontal one, method unchanged.

• Test-point rule

  Substitute any point NOT on the boundary into the original inequality. True: shade that point's side. False: shade the other.

  Use when The origin $(0, 0)$ is easiest, unless the boundary runs through it, when $(1, 0)$ or $(0, 1)$ serves. One substitution settles the whole side, since every point of a side gives the same verdict.

  e.g. $3x + 2y \le 6$ at $(0, 0)$ gives $0 \le 6$, true, so shade the origin's side.

• Corner-point principle

  A linear objective $P = ax + by$ attains its maximum and its minimum at a vertex of the feasible region. If a level line runs parallel to an edge, every point of that edge is optimal, so the corners at its ends tie.

  Use when $a$ and $b$ not both zero. A nonempty, bounded, closed region (every constraint inclusive, $\le$ or $\ge$, so the boundary belongs) GUARANTEES both extremes exist. A strict constraint costs you that guarantee only at its own open edge: an optimum landing there is approached and never reached, while one sitting anywhere else is still attained. On an unbounded region the extreme in the direction it runs off need not exist, so report only the extremes that do exist.

Problem types, step by step

Exam traps

• Trap Forgetting to reverse the symbol after dividing by a negative, turning $-5x > 20$ into $x > -4$.

  Fix The divisor $-5$ is negative, so the direction reverses: $x < -4$. It hides well: the bound comes out right and only the direction is wrong.

• Trap Flipping for the wrong reason: a negative sits somewhere in the problem, you subtracted, or the answer came out negative.

  Fix Only the sign of the number you scale BOTH sides by matters. Solving $2x > -10$ gives $x > -5$ with no flip, because the divisor $2$ is positive.

• Trap Solving something like $-2y > 4x$ for $y$ without flipping, then shading the wrong half-plane.

  Fix Dividing by $-2$ reverses the symbol to $y < -2x$, which moves the shading to the other side. Testing a point in the ORIGINAL inequality avoids the trap entirely.

• Trap Shading a two-variable inequality straight from its symbol, reading $2x - y > 4$ as "greater, so shade above".

  Fix That reflex is only safe once the inequality is solved for $y$, and solving reverses it here: $-y > -2x + 4$ gives $y < 2x - 4$, so the solutions lie BELOW the boundary. Testing $(0, 0)$ gives $0 > 4$, false, which settles the side without the reflex.

• Trap Treating an empty solution set as the number zero, or writing "$x =$" something when the variable cancels.

  Fix No solution means nothing satisfies the inequality, which is not the value $0$; a true leftover means every value does. Neither outcome is a single number.

• Trap Reading $(2, 5)$ as a point in the plane when it is an interval, or the other way round.

  Fix The notations are identical, so the surrounding words decide: an interval is a set of numbers on one axis, a coordinate pair one location in the plane.

• Trap Shading a system's union instead of its overlap, keeping every point that satisfies at least one inequality.

  Fix The feasible region is the intersection: a point belongs only if it satisfies every inequality of the system at once.

• Trap Leaving the inside corner, where two slanted constraints cross, off the vertex list.

  Fix That corner often wins, being the point where both limits are used up completely. Every corner goes on the list before you compare values.