WaywardHouse

13  Inequalities

The range of answers that still work

Your monthly budget for food is £120. There are dozens of ways to spend it — some that work, many that don’t. The question isn’t “what exactly do I spend?” It’s “what combinations are even possible?”

A shelf is rated to hold 50 kg. The boxes you’re stacking weigh 8 kg each. You don’t need the exact weight — you need to know how many boxes you can put on before it becomes unsafe.

A password field requires between 8 and 20 characters. No single length is the answer. Any length in that range works.

These are all inequalities. An equation gives you one answer. An inequality gives you a whole region of answers — and the boundary that separates “works” from “doesn’t work.”

13.1 What the notation is saying

An equation uses \(=\) to say two things are exactly equal. An inequality uses a different symbol to say one thing is bigger or smaller than another.

The four symbols:

Symbol Reads as Example
\(<\) is less than \(3 < 7\)
\(>\) is greater than \(12 > 5\)
\(\leq\) is less than or equal to \(x \leq 10\)
\(\geq\) is greater than or equal to \(n \geq 4\)

The pointy end of the symbol always faces the smaller value. So \(3 < 7\) reads “3 is less than 7” — the small end points at 3, the open end opens toward 7. If you forget which way round it goes, just picture the symbol as an arrow squishing the smaller number.

The difference between \(<\) and \(\leq\) matters. If a shelf holds at most 50 kg, exactly 50 kg is fine — so you’d write \(w \leq 50\). If something must be strictly less than 50, exactly 50 is not fine — so you’d write \(w < 50\).

A compound inequality combines two inequalities into one statement. \(8 \leq n \leq 20\) reads “n is at least 8 and at most 20” — both conditions must hold at once. This is the password length rule: any whole number from 8 to 20 satisfies it.

13.2 The method

Solving an inequality is almost identical to solving an equation. You do the same things: add, subtract, multiply, divide — keeping both sides balanced until the unknown is isolated.

There is one exception. It’s the only genuinely surprising thing in this chapter. It deserves its own section.

13.2.1 The sign-flip rule

When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality reverses.

Here’s why, shown directly. Start with something you know is true:

\[3 < 5\]

Three is less than five. Yes. Now multiply both sides by \(-1\):

\[-3 \quad \text{and} \quad -5\]

Which of these is smaller? \(-5\) is. It’s further left on the number line. So the relationship has flipped:

\[-3 > -5\]

The original said “3 is less than 5.” After multiplying by \(-1\), the true statement is “\(-3\) is greater than \(-5\).”

This isn’t a rule someone invented. It’s what actually happens on the number line when you negate both sides. Multiplying by \(-1\) reflects every number across zero — what was to the left of zero ends up to the right, and the ordering reverses.

The sign-flip rule

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

\[3 < 5 \implies -3 > -5\]

If you forget to flip, your answer will point the wrong way — and every value in your “solution” will actually be wrong.

13.2.2 Solving a linear inequality, step by step

\[2x - 3 < 7\]

Add 3 to both sides (adding is safe — no sign flip):

\[2x < 10\]

Divide both sides by 2 (dividing by a positive number — no flip needed):

\[x < 5\]

The solution is every number less than 5. Not just one number: all of them.

Now compare with one that triggers the flip:

\[-3x + 1 \geq 10\]

Subtract 1 from both sides:

\[-3x \geq 9\]

Divide both sides by \(-3\). This is negative — flip the sign:

\[x \leq -3\]

The solution is every number less than or equal to \(-3\). Check: try \(x = -4\). Then \(-3(-4) + 1 = 12 + 1 = 13\), and \(13 \geq 10\). Yes. Try \(x = 0\). Then \(-3(0) + 1 = 1\), and \(1 \geq 10\). No — and \(x = 0\) is not in our solution set, so that’s correct.

Why solving inequalities works

An inequality is a statement about the relationship between two quantities. Doing the same operation to both sides preserves that relationship — as long as the operation doesn’t reverse the ordering. Addition and subtraction never reverse ordering. Multiplication and division by a positive number preserve ordering. Multiplication or division by a negative number reverses it. That’s the whole story.

13.2.3 Reading the solution on a number line

A number line makes the solution visible. You mark the boundary value and shade the region that satisfies the inequality.

Open circle (\(\circ\)) at the boundary: the boundary value is not included. Use this for strict inequalities (\(<\) and \(>\)).

Closed circle (\(\bullet\)) at the boundary: the boundary value is included. Use this for \(\leq\) and \(\geq\).

For \(x < 5\): draw an open circle at 5, shade to the left.

For \(x \leq 5\): draw a closed circle at 5, shade to the left.

For \(x > -3\): draw an open circle at \(-3\), shade to the right.

For a compound inequality like \(-2 < x \leq 4\): open circle at \(-2\), closed circle at \(4\), shade the segment between them.

Interactive: Number line inequality explorer. Use the slider to set the boundary value. Choose the inequality direction and whether the boundary is included. The shaded region and circle type update live.

13.3 Worked examples

Example 1 — Budgeting. You have £120 to spend on food this month. You’ve already spent £47. You want to buy some items that cost £8 each. How many can you buy without going over budget?

Let \(n\) = number of items.

\[47 + 8n \leq 120\]

Subtract 47 from both sides:

\[8n \leq 73\]

Divide both sides by 8 (positive — no flip):

\[n \leq 9.125\]

Since \(n\) must be a whole number, the answer is \(n \leq 9\). You can buy at most 9 items.

Check: \(47 + 8 \times 9 = 47 + 72 = 119\). Within budget. And \(47 + 8 \times 10 = 127\). Over. Correct.

Example 2 — Shelf load. A wall-mounted shelf is rated to hold at most 50 kg. Each box you place on it weighs 8 kg.

How many boxes can the shelf safely hold?

The shelf load rating tells you how much weight the shelf can carry on top of it. The brackets that mount the shelf to the wall bear their own weight separately — they don’t count against the shelf’s carrying capacity.

Let \(b\) = number of boxes.

\[8b \leq 50\]

\[b \leq 6.25\]

Since \(b\) must be a whole number: at most 6 boxes.

Check: \(8 \times 6 = 48 \leq 50\). ✓ Seven boxes: \(8 \times 7 = 56 > 50\). ✗

Example 3 — Password validation. A website requires passwords to be at least 8 characters long but no longer than 20 characters. A user types a password. The code checks whether its length \(n\) is acceptable.

The valid range is:

\[8 \leq n \leq 20\]

Any value of \(n\) satisfying both conditions is acceptable. \(n = 7\)? Too short. \(n = 8\)? Fine — the lower boundary is included. \(n = 20\)? Fine. \(n = 21\)? Too long.

This is a compound inequality. In code, you’d check both conditions at once: 8 <= n && n <= 20 (or the equivalent in whatever language you’re using). The mathematics and the code are the same structure.

Example 4 — An inequality with a negative coefficient. The temperature in a storage room must be kept strictly below \(5°\text{C}\). The thermostat setting \(t\) (in °C) satisfies:

\[-2t + 14 > 4\]

Subtract 14 from both sides:

\[-2t > -10\]

Divide by \(-2\) — negative, so flip:

\[t < 5\]

The thermostat must be set below 5°C. The sign flip was necessary here: if you forgot it and wrote \(t > 5\), you’d be saying temperatures above 5°C are acceptable, which is exactly wrong.

Interactive: Inequality explorer. Set the coefficients \(a\) and \(b\), choose the constant \(c\) and the inequality direction. The solution is shown algebraically and on a number line. The sign-flip warning lights up when \(a\) is negative.

13.4 Where this goes

Inequalities are where the idea of a constraint enters mathematics. An equation pins down one value. An inequality describes a boundary — everything on one side is feasible, everything on the other isn’t.

The most direct continuation is linear programming — how to find the best point within a region defined by multiple inequalities at once. That’s in the Optimisation chapter in Volume 7. A manufacturer choosing how much of two products to make, subject to labour and material constraints, is solving a linear programme. The feasible region is defined entirely by inequalities like the ones in this chapter.

The second direction is systems of inequalities and their role in applied linear algebra. When a machine learning model classifies data, it’s often drawing a boundary in a high-dimensional space — one side satisfies the classification condition, the other doesn’t. That boundary is expressed as an inequality. The linear equations you learned in the previous chapter define the boundary; the inequality says which side is which. That thread runs from here through linear algebra (Volume 4) all the way to optimisation and machine learning.

Where this shows up

  • A financial analyst building a budget model sets upper and lower bounds on each line of spending — those bounds are inequalities.
  • A structural engineer checks that the load on every beam stays within its rated capacity — an inequality checked against the design at every point.
  • A software developer writing input validation checks that every user-entered value lies in an acceptable range — a compound inequality executed in code.
  • A supply chain planner ensures stock levels stay between a minimum (to avoid running out) and a maximum (to avoid overflow) — two inequalities, applied simultaneously.

13.5 Exercises

Each problem has a clean answer. The work is in translating the situation into the notation before you solve it.

  1. You can spend at most £50 per month on streaming services. You already pay £14.97 for two services. How many additional services at £9.99 per month can you add and still stay within your budget?
  1. A lift (elevator) has a maximum capacity of 630 kg. A group of adults with an average weight of 78 kg wants to use it. What is the largest number of people who can ride together safely?
  1. Solve for \(x\) and mark the solution on a number line: \[-4x + 5 > 17\] Explain in one sentence what happens to the inequality sign and why.
  1. A password must be at least 8 characters. It must also be fewer than 3 times the minimum length. Write this as a compound inequality for the password length \(n\), then state the range of valid lengths.
  1. Solve the compound inequality and state which whole numbers satisfy it: \[1 < 3x - 2 \leq 13\]
  1. A phone plan charges £0.08 per minute after a 200-minute monthly allowance. You want your total bill to be under £25. The base plan costs £12. How many additional minutes can you use?
  1. A company’s safety guideline says a storage shelf should never hold more than 75% of its rated maximum. The shelf is rated to 80 kg. Boxes weigh 6 kg each. What is the maximum number of boxes you can store on this shelf within the guideline?