How Many Hours of Tuition Fit Your Budget? One Inequality Answers It

Many real conditions are limits rather than exact values: at least, at most, more than, less than. An inequality states such a condition using $<$, $>$, $\le$ or $\ge$.

Solving an inequality gives a whole range of acceptable values, usually written as an interval. This chapter covers types of inequalities, solving them, showing solutions on a number line, systems of inequalities and word problems.

**Strict inequalities use $<$ or $>$ and exclude the boundary value, slack inequalities use $\le$ or $\ge$ and include it, and an inequality is linear in one variable if it can be written as $ax+b<0$ or similar, with $a\ne 0$.

Strict or slack?**

- $x<5$ — strict; $5$ itself does not satisfy it
- $x\ge 18$ — slack; $18$ does satisfy it

Linear in one variable or not?

- $3x-5<7$ — linear in one variable
- $x^2>4$ — not linear, because of $x^2$
- $2x+3y\le 6$ — linear, but in two variables
- $\frac{1}{x}>2$ — not linear, because $x$ is in a denominator

An everyday example. **A rule that voters must be at least $18$ years old** is the slack inequality $x\ge 18$, while a scheme giving free entry to children under $5$ is the strict inequality $x<5$.

The substance. **The difference between $<$ and $\le$ decides whether the end point belongs to the answer**, which shows up as a round or square bracket later.

Collect the variable on one side exactly as in an equation; adding, subtracting, or multiplying and dividing by positive numbers keep the sign, but multiplying or dividing by a negative number reverses it.

Worked example 1. Solve $4x+3<6x+7$.

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

Worked example 2. Solve $\frac{2x-1}{3}\ge \frac{3x-2}{4}-\frac{2-x}{5}$. Multiply by $60$:

$$20(2x-1)\ge 15(3x-2)-12(2-x)$$
$$40x-20\ge 57x-54-17x\ge -34x\le 2$$

Worked example 3 — depends on the number set. Solve $30x<200$.

$$x<\frac{20}{3}\approx 6.67$$

- **For $x\in \mathbb{N}$**: $\{1,2,3,4,5,6\}$
- **For $x\in \mathbb{R}$**: all real numbers less than $\frac{20}{3}$

An everyday example. Working out how many ₹30 notebooks fit in a ₹200 budget is exactly the third example, with a whole-number answer of at most $6$.

The substance. **Never multiply both sides by an expression like $x-1$** whose sign is unknown — it may or may not reverse the inequality.

Mark each boundary with a hollow circle if excluded or a filled circle if included, shade the allowed side, and write the result as an interval; for a system, keep only the values that satisfy every inequality.

Single inequalities.

- $x>-2$: hollow circle at $-2$, shade right; interval $(-2,\infty )$
- $x\le 2$: filled circle at $2$, shade left; interval $(-\infty ,2]$

Worked example 1 — a system. Solve $3x-7<5+x$ and $11-5x\le 1$.

$$2x<12\Rightarrow x<6-5x\le -10\Rightarrow x\ge 2$$

Both together: $[2,6)$.

Worked example 2 — a double inequality. Solve $-5\le \frac{5-3x}{2}\le 8$.

$$-10\le 5-3x\le 16-15\le -3x\le 11-\frac{11}{3}\le x\le 5$$

Interval: $\left[-\frac{11}{3},5\right]$.

An everyday example. **Many vaccines must be stored between $2^{\circ }$C and $8^{\circ }$C inclusive**, the interval $[2,8]$.

The boundary case. A system can have no solution: $x>5$ and $x<2$ share no values, so the solution set is $\varnothing$.

**Choose a variable for the unknown quantity, translate phrases such as at least and at most into $\ge$ and $\le$, solve, and then check that the answer makes sense in the situation.

Worked example 1 — tuition.** A student has ₹5000. A coaching centre charges ₹800 to register and ₹350 per hour. How many hours can the student afford?

$$800+350h\le 5000350h\le 4200h\le 12$$

**At most $12$ hours.

Worked example 2 — marks.** Ravi scored $70$ and $75$ in two unit tests. What must he score in the third to average at least $60$?

$$\frac{70+75+x}{3}\ge 60145+x\ge 180x\ge 35$$

Worked example 3 — consecutive odd numbers. Both are greater than $10$ and their sum is less than $40$.

$$x>10x+(x+2)<40\Rightarrow x<19$$

The pairs are $(11,13)$, $(13,15)$, $(15,17)$ and $(17,19)$.

Worked example 4 — a solution. How many litres of $2\%$ acid must be added to $640$ L of $8\%$ acid so that the mixture is between $4\%$ and $6\%$?

$$0.04(640+x)<51.2+0.02x<0.06(640+x)320<x<1280$$

An everyday example. Families planning monthly spending set limits like these whenever a budget is fixed.

The substance. Interpret the answer: hours and numbers of objects must be whole numbers within the interval.