Why Multiplying by a Negative Number Flips an Inequality Sign

A linear inequation compares two expressions using $<$, $>$, $\le$ or $\ge$, such as $3x-5\le 7$. Instead of one answer, it usually has many values that work, called the solution set. The set of values $x$ is allowed to take is the replacement set.

Rules for solving:

- Adding or subtracting the same number on both sides keeps the sign
- Multiplying or dividing by a positive number keeps the sign
- Multiplying or dividing by a negative number reverses the sign

Why the flip happens: $2<5$, but $-2>-5$.

This chapter covers solution sets, number lines, combined inequations and word problems.

Solve exactly as for an equation, reversing the sign if you multiply or divide by a negative number, then list the values from the replacement set that satisfy the result.

Worked example 1. Solve $3x-5\le 7$.

$$3x\le 12x\le 4$$

- $x\in N$: $\{1,2,3,4\}$
- $x\in W$: $\{0,1,2,3,4\}$
- $x\in Z$: $\{\dots ,-2,-1,0,1,2,3,4\}$
- $x\in R$: $\{x:x\in R,x\le 4\}$

Worked example 2. Solve $-2x+3>11$.

$$-2x>8x<-4\text{(sign reversed on dividing by }-2\text{)}$$

- $x\in N$: **no natural number is less than $-4$**, so the solution set is the empty set $\varnothing$
- $x\in Z$: $\{\dots ,-7,-6,-5\}$

An everyday example. **If a shop allows at most $4$ packets of a sale item per customer**, the possible numbers bought are $x\le 4$ with $x\in W$, since buying zero is also allowed.

The substance. The same inequation can have different solution sets depending on the replacement set — and sometimes no solution at all.

For real numbers, shade the line between the endpoints, with a solid dot where the endpoint is included and a hollow circle where it is not; for natural numbers, whole numbers or integers, mark only the separate points.

Symbols to use:

- Solid dot — endpoint included, for $\le$ or $\ge$
- Hollow circle — endpoint excluded, for $<$ or $>$
- Thick line with an arrow — all real values continue in that direction

Worked example. Show $-3<x\le 2$.

- **For $x\in R$: draw a hollow circle at $-3$, a solid dot at $2$, and shade the line between them
- For $x\in Z$: mark separate dots** at $-2,-1,0,1,2$ only

Worked example. Show $x\ge 1$, $x\in R$: a **solid dot at $1$ with a thick line and arrow going to the right.

An everyday example. A road sign showing a maximum speed of $50$ km/h** allows every speed up to and including $50$.

The substance. A hollow circle where a solid dot belongs changes the answer.

**Carry out the same operation on all three parts at once until $x$ stands alone in the middle, reversing both signs if you divide by a negative number.

Worked example 1.** Solve $-3<2x-1\le 5$, $x\in R$.

$$-3+1<2x\le 5+1-2<2x\le 6-1<x\le 3$$

Solution set: $\{x:x\in R,-1<x\le 3\}$. For $x\in Z$: $\{0,1,2,3\}$.

Worked example 2 — negative coefficient. Solve $4\le 7-3x<13$, $x\in W$.

$$-3\le -3x<61\ge x>-2-2<x\le 1$$

Solution set for $x\in W$: $\{0,1\}$.

Worked example 3 — two separate inequations. Find $x\in R$ with $2x+5\ge 1$ and $3x-4<8$.

$$2x\ge -4\Rightarrow x\ge -23x<12\Rightarrow x<4\text{so }-2\le x<4$$

An everyday example. **A school awarding grade B for marks from $60$ up to but not including $75$** is describing $60\le m<75$.

The substance. **Dividing by $-3$ reverses both signs.**

Solve, then pick the largest or smallest allowed value; in word problems, turn at most and at least into inequality signs first.

Worked example 1. Find the greatest integer $x$ with $5x-3<2x+12$.

$$3x<15x<5\text{greatest integer}=4$$

Worked example 2. Find the greatest natural number $x$ with $\frac{x}{3}-1>\frac{x-5}{2}$.

$$2x-6>3x-15-x>-9x<9\text{greatest natural number}=8$$

Worked example 3 — money. Riya has ₹500. She buys one notebook for ₹45 and some pens at ₹35 each. How many pens can she buy at most?

$$35x+45\le 50035x\le 455x\le 13$$

**At most $13$ pens.

Worked example 4 — marks.** Arjun scored $72$ and $65$ in two tests out of $100$. What must he score in the third to average at least $75$?

$$\frac{72+65+x}{3}\ge 75137+x\ge 225x\ge 88$$

Since a test is out of $100$: $88\le x\le 100$.

An everyday example. Budgeting pocket money is solving an inequation.

The substance. Always check the context, such as whole numbers and maximum marks.