Secondary 1 Equations and Inequalities Worksheet

All 4 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Representing a Solution Set

   A food truck's freezer is working properly as long as its temperature $t$, in degrees Celsius, satisfies $t\le -3$.

   1. Describe the solution set in words.
   2. Draw a number line from $-8$ to $2$ and shade the solution set on it. Use a filled dot at $-3$ to show that $-3$ belongs to the set.
   3. The thermometer read $-2.5{}^{\circ }\text{C}$ on Monday, $-3{}^{\circ }\text{C}$ on Tuesday and $-6{}^{\circ }\text{C}$ on Wednesday. On which days was the freezer working properly?
2. Q2Solving Equations and Inequalities

   Solve each of the following. At every step, state which operation you are undoing.

   1. $4x+9=41$
   2. $\frac{n}{3}-5=7$
   3. $2m+13\le 27$, where $m$ is a whole number. Give every value of $m$ that works.
3. Q3Translating a Statement Into an Equation or an Inequality

   For each statement, choose a letter for the unknown quantity, say clearly what it represents, and translate the statement into an equation or an inequality. Do not solve, except where part e) asks you to.

   1. A bubble-tea shop sells every large cup at $6. Inès spent $42 on large cups.
   2. A school bus may carry no more than $48$ passengers.
   3. Five more than triple a number gives $26$.
   4. After paying $15 for a movie ticket, Théo has less than $20 left from his savings.
   5. Solve the equation you wrote in part c).
4. Q4Synthesis — drawing on several sheets in this topic

   A community centre rents its gym for a fixed booking fee of $25, plus $18 for each hour. A hockey club has $160 to spend and books the gym for a whole number of hours $h$, with $h$ at least $1$.

   1. Translate “the total cost must not go over $160” into an inequality in $h$.
   2. Solve your inequality.
   3. List every number of hours the club can book, and explain how you would show this solution set on a number line.

The 4 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Solving is undoing, in reverse order

An equation is a sentence claiming two things are equal. Solving it means peeling the operations off the unknown one at a time — and Q2 asks you to state which operation you are undoing at each step, which is worth doing even when nobody asks.

The order is the part to get right. Whatever was done last to the unknown is undone first. In 4x + 9 = 41, the 9 was added after the multiplication, so the 9 comes off first and the 4 is divided out second. Going the other way — dividing everything by 4 while the 9 is still there — is not wrong, but it produces fractions where there were none, and at this level that is how an answer gets lost.

An inequality is solved the same way, and answered differently (Q2c)

2m + 13 ≤ 27 is undone exactly like an equation: take off the 13, divide by 2, and you reach m ≤ 7. What changes is what counts as a finished answer. An equation has one number; an inequality has a whole collection of them, and the question tells you which collection by telling you what m is.

Here m is a whole number, so the answer is the list 0, 1, 2, 3, 4, 5, 6, 7 — not the single value 7, and not every number below 7. Reading that condition out of the question is part of the question.

Translating: the direction of a subtraction is not optional (Q3)

Q3 is the translation question, and it insists on a step people skip: say what the letter stands for before you use it. “Let c be the number of large cups” is not decoration. Without it, 6c = 42 is an equation about nothing, and a marker cannot tell whether you meant cups or dollars.

Representing a solution set: dots or a band (Q1)

Q1 puts a freezer at t ≤ −3 and asks for the set in words, on a number line, and applied to three real readings. The number line is the part that goes wrong, and it goes wrong in two different ways.

Q1(c) is the reality check, and the trap in it is the negative numbers. A reading of −2.5 fails the condition t ≤ −3, because −2.5 is warmer than −3 and therefore larger. Further from zero on the negative side means smaller, and that reversal catches people every year.

Q4: the whole method on one problem

The last core question rents a gym at a fixed booking fee plus an hourly rate, with a budget to stay inside. It translates into 25 + 18h ≤ 160, solves to h ≤ 7.5, and then asks the question that matters: what does a fractional answer mean when the thing being counted is whole hours?

It means the answer is 7, and it means the number line carries seven separate dots rather than a shaded band. Coming back from the algebra to the situation is the step the question is really testing — and it is the step that solving practice alone never teaches.

Getting the most out of it

Name the unknown in a full sentence

“Let h be the number of hours booked.” Write it before the equation, every time. It costs one line, it forces you to decide what you are actually solving for, and it is the difference between an answer of 7 and an answer of 7 hours.

Undo in reverse order, one operation per line

Whatever happened last to the unknown comes off first. Write each step on its own line with the operation named beside it. A solution written as one long chain is impossible to check, and almost impossible to earn part marks on.

Ask what the letter counts before you draw the number line

Students, hours and tickets get dots. Temperatures, masses and lengths get a shaded band. The inequality looks the same either way, so the only place the answer can come from is the situation described in the question.

Common questions