Secondary 3 Equations and Inequalities Worksheet

All 7 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. Q1Algebra - Equations and Inequalities

   Translate each statement into an equation or an inequality using the variable named, say which of the two you wrote, then solve it.

   1. A skatepark charges a $4 entry fee plus $2.50 for each hour on the ramps. Léa paid $14 in all. Let $h$ be her number of hours.
   2. A minibus may carry at most $22$ passengers. Seven adults are already seated and $n$ students still have to board.
   3. Three times a number, decreased by $8$, is more than $25$. Let $n$ be the number.

   Finally, say which of your three answers is a single value and which is a whole collection of values.

2. Q2General Methods for Solving Equations

   Solve each equation, then verify your solution in the original equation.

   1. $6x-11=2x+21$
   2. ${\displaystyle \frac{2x+1}{3}}=7$
   3. $5(2x-3)-4x=27$
3. Q3Representing a Solution Set

   Answer each part.

   1. Solve $2x-9\le 5$ over the real numbers. Give the solution set in set-builder notation and in interval notation, and say whether the endpoint on a number line is a solid or an open point.
   2. Write out, as a list between braces, the set $\{x\in \mathbb{Z}\mid -3\le x<4\}$.
   3. A ride at the fair admits anyone whose height is at least $130$ cm and less than $195$ cm. Write the set of admissible heights, in centimetres, in interval notation and in set-builder notation.
4. Q4Solving Algebraic Inequalities

   Solve each inequality over the real numbers and give each solution set in interval notation.

   1. $5x+8<3x+20$
   2. $-2x+7\ge 15$
   3. ${\displaystyle \frac{x}{4}}-3>1$

   Then state which part forced you to reverse the inequality symbol, and why.

5. Q5Solving Equations and Inequalities

   A community garden charges each member a $90 seasonal plot fee plus $12 for every bag of compost taken. Let $b$ be the number of bags.

   1. Write the rule giving the total cost, in dollars.
   2. Rosalie's bill came to exactly $450. Write and solve an equation to find how many bags she took.
   3. Marc will not spend more than $450. Write and solve an inequality, and state how many bags he may take.
6. Q6Solving First-Degree Equations and Inequalities

   Solve each of the following. For each one, state how many solutions it has.

   1. $4(x-3)=2(x+5)$
   2. $3(x+2)>5x-4$, over the real numbers, with the answer in interval notation
7. Q7Synthesis — drawing on several sheets in this topic

   A hockey tournament charges each team a $145 registration fee plus $35 for every player on its roster.

   1. Write the rule giving the total cost $C$ for a team of $n$ players.
   2. The Rimouski team paid $600. Write and solve an equation to find its number of players, and verify your answer.
   3. A second team's budget is at most $810. Write and solve an inequality, give the solution set in interval notation, then state how many players the team may actually register and explain why the interval on its own is not the final answer.

The 7 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.

The one difference between an equation and an inequality

Q1 asks you to translate three situations, say which of the two you wrote, and then say which answers are a single value and which are a whole collection of values. That is the entire idea of the topic, met before any technique. An entry fee plus an hourly rate coming to exactly $14 pins down one number; a minibus that carries at most 22 passengers does not.

The steps are the same — the same things may be added, subtracted, multiplied and divided on both sides — with one exception, and it is the one place this topic reliably goes wrong.

Verifying is part of the answer, not a nicety

Q2 says solve, then verify, and it means substitute your value back into the original equation and show both sides landing on the same number. Two of its three parts are built to punish a rushed step — a fraction over a whole expression, and a bracket that has to be distributed before anything else can happen — and both are caught instantly by a check that takes one line.

Substituting 7 back: 5(14 − 3) − 28 = 55 − 28 = 27. Notice that the check uses the original equation, not the tidied-up version — checking against your own third line confirms only that you can copy.

Three ways to write the same solution set

Q3 is the sheet's real novelty. One answer, three notations, and each one is a place a mark can be lost on its own.

Context can shrink an answer the algebra cannot

The synthesis question is the clearest case on the sheet. A team's budget gives an inequality whose solution set, in interval notation, is a stretch of real numbers — and then the question asks how many players the team may actually register, and points out that the interval on its own is not the final answer. Players come in whole numbers, and the interval does not know that.

The same thing decides the challenge question about bracelets, where a treasurer produces the number that makes the profit exactly $200 and calls it the answer to a question asking for more than $200. A strict inequality excludes its own boundary. When the variable also has to be a whole number, the honest answer is the first whole number strictly past it, and saying which whole number and why is what the question is marked on.

Two conditions at once

The last challenge question puts two orders at a school fair side by side — three poutines and two lemonades for one total, one poutine and four lemonades for another — and both statements have to hold at the same time. That is a system of two first-degree equations in two unknowns, and it is the natural place this topic ends up: one equation cannot pin down two prices, and the second order is what makes the pair solvable.

It also contains the misreading that motivates the whole idea. Dividing the second total by four and calling the result the price of a lemonade quietly assumes the poutine in that order was free. Each equation constrains the pair of prices; neither one on its own determines either price.

Getting the most out of it

Write down which one you are solving before you start

Equation or inequality — say it out loud. Everything downstream depends on it: whether the answer is a number or a set, whether you owe a solution set at the end, and whether a division could turn the symbol around.

Substitute back into the original

Not into your second line. The check is only worth doing against the equation you were given, because that is where a distribution error or a dropped sign lives. For an inequality, test a number from inside your solution set and one from outside it; if both behave, the boundary is in the right place and pointing the right way.

Draw the number line even when nobody asked

It takes five seconds and it is the fastest way to catch a symbol that should have flipped. If your algebra says one thing and the picture says the other, the picture is usually right.

Finish with a sentence about the situation

An interval is a mathematical answer; "the team can register at most 19 players" is the answer to the question that was asked. The last step of a word problem is turning the set back into people, bags, bracelets or dollars, and checking that the number you are about to give could exist.

Common questions