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2.4 — Inequalities and Systems of Equations
Most real questions do not ask for an exact value. They ask for a range: how many units must we sell to at least break even, what dosage is safe, how fast can the server run before latency exceeds the limit. And most real problems involve several unknowns tangled together rather than one.
This chapter handles both — the mathematics of "at least" and the mathematics of several conditions at once.
1. Inequalities: everything works the same except one thing
The symbols:
- a \lt b — a is less than b; on the number line of Chapter 1.2, a sits to the left.
- a \le b — less than or equal to.
- a \gt b, a \ge b — the mirror images.
Solving an inequality uses the same balance moves as an equation. You may add or subtract anything from both sides freely, and multiply or divide by any positive number freely.
The one rule that is different
Multiplying or dividing both sides by a negative number reverses the inequality.
Start with something true: 3 \lt 5. Multiply both sides by -1: is -3 \lt -5? No. On the number line, -3 sits to the right of -5, so -3 \gt -5. The direction flipped.
Why: multiplying by a negative reflects the whole number line through zero, and a reflection swaps left and right. Order is a statement about position, and the reflection reverses positions.
Worked example. Solve -3x + 7 \le 1.
-3x \le -6
Divide by -3 — negative, so flip:
x \ge 2
Check with a value: x = 5 gives -15 + 7 = -8, and -8 \le 1. True. Check the other side: x = 0 gives 7, and 7 \le 1 is false. Correct.
Notation for the answer. The solution is a set, and there are three common ways to write it. As an inequality: x \ge 2. In interval notation: [2, \infty), where a square bracket includes the endpoint and a round bracket excludes it. Infinity always gets a round bracket, because it is not a number you can reach. As a picture: a filled dot at 2 and a ray heading right.
2. Quadratic inequalities: solve, then check the sign in each region
x^2 - 5x + 6 \gt 0 cannot be attacked by moving things around, because x^2 and x do not combine. The method is different and it is a method you will use for the rest of the book.
Step 1. Get zero on one side. Already done.
Step 2. Find where the expression equals zero — its roots. Factoring: (x-2)(x-3) = 0, so x = 2 and x = 3.
Step 3. Those roots cut the number line into three regions. Inside each region the expression cannot change sign, because to change sign a continuous expression must pass through zero, and there are no zeros inside a region. So test one convenient point per region.
- x = 0: (0-2)(0-3) = 6 \gt 0. Positive.
- x = 2.5: (0.5)(-0.5) = -0.25 \lt 0. Negative.
- x = 4: (2)(1) = 2 \gt 0. Positive.
Answer: x \lt 2 or x \gt 3, written (-\infty, 2) \cup (3, \infty), where \cup means "or" — the union of the two intervals.
This makes sense against Chapter 2.3: the parabola opens upward and crosses at 2 and 3, so it is above the axis outside the roots and below between them.
The "test a point in each region" method works for any expression that is continuous — polynomials, and rational functions if you also mark where the denominator vanishes. The underlying principle is the Intermediate Value Theorem of Chapter 5.1, which says a continuous function cannot get from positive to negative without passing through zero.
3. Inequalities with absolute value
Recall from Chapter 1.2 that |x| is distance from zero. Read the inequality as a sentence about distance and the answer is immediate.
|x| \lt 5 says "the distance from zero is less than 5", so x is within 5 of zero: -5 \lt x \lt 5.
|x| \gt 5 says "the distance is more than 5", so x is outside that band: x \lt -5 or x \gt 5.
Notice that "less than" gives one interval joined by and, while "greater than" gives two intervals joined by or. That is the part people get wrong, and reading it as distance prevents it.
Worked example. |2x - 7| \le 3.
The distance from 2x to 7 is at most 3, so 2x is between 4 and 10:
-3 \le 2x - 7 \le 3 \quad\Rightarrow\quad 4 \le 2x \le 10 \quad\Rightarrow\quad 2 \le x \le 5
This shape appears constantly in engineering as a tolerance: a component specified as 50 \pm 0.2\ \text{mm} is exactly the statement |d - 50| \le 0.2. And in Chapter 5.1 the definition of a limit is written entirely with two such inequalities.
4. Systems of equations: two unknowns, two facts
One equation with two unknowns has infinitely many solutions. x + y = 10 is satisfied by (0,10), (3,7), (5.5, 4.5) and endlessly more; plotted, they form a line.
To pin down two unknowns you need two independent facts. Geometrically: two lines, and the solution is where they cross.
\begin{cases} 2x + 3y = 12 \\ x - y = 1 \end{cases}
Method 1: substitution
Solve the easier equation for one variable and put it into the other.
From the second: x = y + 1. Substitute into the first:
2(y+1) + 3y = 12 \quad\Rightarrow\quad 5y + 2 = 12 \quad\Rightarrow\quad y = 2
Then x = 3. Check both: 2(3)+3(2) = 12 ✓ and 3 - 2 = 1 ✓.
Method 2: elimination
Scale the equations so one variable cancels when you add or subtract.
Multiply the second by 3: 3x - 3y = 3. Add it to the first:
(2x + 3y) + (3x - 3y) = 12 + 3 \quad\Rightarrow\quad 5x = 15 \quad\Rightarrow\quad x = 3
Then y = 2. Same answer.
Which to use. Substitution when one variable already has a coefficient of 1. Elimination otherwise, and always for three or more equations — it is the method that scales, and Chapter 4.3 turns it into a mechanical procedure called Gaussian elimination that a computer runs on systems with thousands of unknowns.
The three possible outcomes
Two lines in a plane can do exactly three things, and each corresponds to an algebraic signature.
Cross at one point — one solution. The normal case.
Be parallel and distinct — no solution. The algebra tells you by producing a false statement: try x + y = 5 with x + y = 8, and elimination gives 0 = 3. The system is inconsistent.
Be the same line — infinitely many solutions. The algebra produces a trivially true statement: x + y = 5 with 2x + 2y = 10 gives 0 = 0. The second equation told you nothing new; the equations are dependent.
Recognising which case you are in from the algebra alone matters, because when a computer solves a large system it has to report exactly this, and because "no solution" and "infinitely many solutions" mean very different things about the problem you were modelling. Chapter 4.3 gives the general test.
5. Three unknowns, and beyond
Same idea, more bookkeeping. Use elimination to remove one variable from two of the equations, leaving a two-by-two system you already know how to solve, then substitute backwards.
\begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x + 2y - z = 2 \end{cases}
Add equations 1 and 3 to kill z: 2x + 3y = 8. Add equations 2 and 3 to kill z again: 3x + y = 5.
Now a two-by-two system. From the second, y = 5 - 3x. Substitute:
2x + 3(5-3x) = 8 \quad\Rightarrow\quad 2x + 15 - 9x = 8 \quad\Rightarrow\quad -7x = -7 \quad\Rightarrow\quad x = 1
Then y = 2, and from equation 1, z = 3. Check all three: 1+2+3=6 ✓, 2-2+3=3 ✓, 1+4-3=2 ✓.
Geometrically these are three planes in space, and the solution is the single point all three share. They might instead meet in a line, or not meet at all — the same three outcomes.
By four or five unknowns the arithmetic becomes unmanageable by hand, and that is the honest motivation for Part 4. Linear algebra is not a new subject; it is this chapter, organised so a machine can do it.
6. Systems of inequalities, and the shape of the answer
Replace the equals signs with inequalities and the solution stops being a point and becomes a region.
\begin{cases} x + y \le 10 \\ 2x + y \le 16 \\ x \ge 0,\; y \ge 0 \end{cases}
Each inequality cuts the plane in half. The solution is where all four half-planes overlap — a polygon, called the feasible region.
This is exactly the shape of a real planning problem. Suppose a workshop makes chairs (x) and tables (y). Wood available limits x + y \le 10; labour hours limit 2x + y \le 16; you cannot make negative furniture, hence the last two. Every point in the polygon is a production plan you could actually carry out.
Now add a goal: profit is ₹300 per chair and ₹500 per table, so maximise 300x + 500y.
The key fact, and it is not obvious: the maximum always occurs at a corner of the polygon. The reason is that the profit expression is linear, so its contour lines are parallel straight lines; sliding them across the region, the last point of contact before leaving is always a vertex (or, in a tie, a whole edge including two vertices). So you never need to test the interior — check the corners and you are done.
Corners here are (0,0), (8,0), (6,4) and (0,10). Profits: 0, 2400, 3800, 5000. Best plan: 10 tables, no chairs, ₹5,000.
This is linear programming, and Chapter 10.5 develops it properly. It is the technique behind airline scheduling, refinery blending, delivery routing and portfolio construction, and it is arguably the most economically valuable piece of mathematics invented in the twentieth century.
7. Two inequalities worth knowing by name
The triangle inequality.
|a + b| \le |a| + |b|
In words: the size of a sum is never more than the sum of the sizes. It is called the triangle inequality because in Chapter 3.2's geometry it says one side of a triangle is never longer than the other two combined — going direct is never longer than going via a detour. Equality holds only when a and b point the same way. This single statement reappears as the defining axiom of "distance" in every space mathematics builds, all the way to Chapter 4.1's vectors and Volume I's similarity measures.
The arithmetic-geometric mean inequality, for non-negative numbers:
\frac{a+b}{2} \ge \sqrt{ab}
The ordinary average is never less than the geometric mean, with equality only when a = b. Proof in one line: (\sqrt a - \sqrt b)^2 \ge 0, since a square is never negative. Expand it: a - 2\sqrt{ab} + b \ge 0, rearrange, divide by 2.
Its practical meaning: for a fixed perimeter, the square encloses the most area. Set a and b as the sides of a rectangle; the left side is fixed by the perimeter, the right side is the area's square root, and it is largest when the sides are equal. That is a cleaner proof of the fencing-type problem in Chapter 2.3 than completing the square, and it generalises to any number of dimensions — which is why bubbles are spheres.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Inequalities
\text{If } a < b \text{ then } a + c < b + c \text{ for any } c
\text{If } a < b \text{ and } c > 0 \text{ then } ac < bc
\text{If } a < b \text{ and } c < 0 \text{ then } ac > bc \quad \text{— the direction flips}
Why multiplying by a negative flips it. Take 2 < 5 and multiply by -1. On the number line, multiplying by -1 reflects every point through zero. The reflection turns 2 into -2 and 5 into -5, and -5 sits to the left of -2. Reflection reverses order, so the inequality sign must reverse with it. Forgetting this is the single most common error in the whole of algebra.
Absolute value.
|x| < a \;\Leftrightarrow\; -a < x < a, \qquad |x| > a \;\Leftrightarrow\; x < -a \text{ or } x > a
Read |x| as the distance from x to zero, and both lines say something obvious about distance: being within a of zero means being in the band from -a to a; being further than a away means being outside it, on one side or the other.
The triangle inequality.
|a + b| \le |a| + |b|
Two steps of size |a| and |b| get you at most |a|+|b| from the start, and less if they partly oppose each other. Equality holds exactly when a and b point the same way. The name comes from the geometric version in 4.1: one side of a triangle is never longer than the other two combined.
8. Where this shows up in your life
Every "you need at least X" calculation. Break-even points, minimum order quantities, qualifying thresholds.
Every engineering tolerance. 50 \pm 0.2\ \text{mm} is an absolute-value inequality, and the whole discipline of tolerance stack-up is adding them.
Every scheduling or allocation problem solved by software — flight crews, delivery routes, cloud instance placement, hospital rosters. Feasible region, corners, linear programming.
Every service level objective in software. "99% of requests under 200 ms" is a statement about a distribution satisfying an inequality; Volume I, 10.9 handles the engineering side.
Every constraint in a spreadsheet Solver. Excel's Solver is a linear programming engine, and the constraints you type are literally the system of Section 6.
One case remains unsolved from Chapter 2.3: a quadratic with a negative discriminant, where the formula asks for the square root of a negative number. The next chapter takes that request seriously.