Secondary 2 Equations and Inequalities Worksheet

5 of the 6 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 5 of the 6 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1General Methods for Solving Equations

   Solve each equation using the method named, then verify your solution by substituting it back into the original equation.

   1. Balancing: $4x+7=2x+19$
   2. Inverse operations: ${\displaystyle \frac{3x}{5}}-4=8$
   3. Hidden terms (cover-up): $7(x-3)=56$
2. Q2Representing a Solution Set

   This question is built around a diagram or a table of values. Open it in the PDF.

3. Q3Solving Equations and Inequalities

   Two equations (or inequalities) are equivalent when they have exactly the same solution set. For each pair, decide whether the two are equivalent. Name the transformation used, or show the two solution sets.

   1. $3x=27$ and $3x+8=35$
   2. $x-4=6$ and $5(x-4)=30$
   3. $-2x<8$ and $x<-4$
4. Q4Solving First-Degree Equations and Inequalities

   Solve each equation or inequality. For the inequality, state the solution set in interval notation.

   1. $5x-8=3x+10$
   2. $2(x+4)-3x=11$
   3. ${\displaystyle \frac{x}{3}}+2={\displaystyle \frac{x-4}{2}}$
   4. $7-4x\ge 2x+25$
5. Q5Translating a Statement Into an Equation or an Inequality

   Translate each statement into one equation or one inequality, using the variable given. Do not solve.

   1. A bubble-tea shop sold three times as many taro drinks as mango drinks, and $96$ drinks in all. Let $m$ be the number of mango drinks.
   2. Zach is $5$ years younger than Amel. In $4$ years, their two ages will add up to $41$. Let $a$ be Amel's age today.
   3. The climbing room holds no more than $24$ people at a time, and $9$ climbers are already inside. Let $p$ be the number who may still enter.
   4. A number decreased by $7$ is at least twice the number. Let $n$ be the number.
6. Q6Synthesis — drawing on several sheets in this topic

   A student council orders reusable water bottles printed with the school logo. The printer charges $60 to set up the design, plus $4.50 per bottle. The council sells each bottle for $12 and expects to sell every one it orders. Let $n$ be the number of bottles.

   1. Translate into an inequality the statement “the money taken in is at least as large as the amount paid to the printer”, then solve it.
   2. Write the solution set in set-builder notation, choosing the number set that fits the situation, and justify your choice.
   3. Verify the endpoint of your solution set in the original inequality.

The 6 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.

Three methods, and how to tell which one the equation wants

Students who know only one way of solving are slow, and slow is where sign errors come from. Each of these is quickest on a particular shape, and one of the questions names the method it wants precisely so you have to recognise the shape.

Verifying is part of the method, not an optional extra

Substituting your answer back into the original equation is the only step that can catch an error made anywhere in the working, and it takes one line. The sheet asks for it explicitly on the first question and then dramatises what skipping it costs: a solution that "felt right" produces 27 where the equation says 35, and the check says so instantly.

Substitute into the original, never into a line you wrote yourself — a check against your own third line will happily confirm a mistake made in your second.

Equivalent equations: what you are allowed to do

Two equations are equivalent when they have exactly the same solutions. Adding or subtracting the same number on both sides preserves the solutions; so does multiplying or dividing both sides by the same non-zero number. That is what licenses every step you take, and one question here asks you to name the transformation rather than just to solve.

Inequalities: one rule, and it is the only one

Everything you do to an equation you may do to an inequality, with a single exception: multiplying or dividing both sides by a negative number reverses the sense of the sign.

Adding and subtracting never reverse anything, whatever signs are involved — a point worth fixing in your head, because most students who over-apply the rule do it while moving a term across. And there is a way round it entirely: move the variable term to the other side so its coefficient ends up positive, and you never divide by a negative at all. One of the questions here works through that alternative and asks whether the rule is still needed. It is: you will meet solutions written both ways, and you have to be able to read them.

Writing the solution set, and choosing the right number set

A solved inequality is not yet an answer; the answer is the set of numbers that work, written in a way someone else can read. Three notations say the same thing:

In the Québec convention the bracket turns outward at an endpoint that is excluded, so ]−1, +∞[ starts just after −1, and infinity always takes an outward bracket because it is never reached. On a number line the same distinction is a solid point for an included endpoint and an open circle for an excluded one.

Rounding a context answer: up or down is decided by the situation

Sixteen point eight driveways is not a number of driveways. Whether you round up or down is not a rule about decimals — it is a question about the situation. A budget that must not be exceeded rounds down, because the seventeenth driveway costs more than there is. A number of packs that must feed everyone rounds up, because eight and a bit packs leaves people without. Say which of the two you are in, in a clause, and the mark follows.

Translating a statement

Name the variable first, in writing, and say what it stands for including its unit. Everything else in the sentence is then expressed in terms of it: if Amel is a years old and Zach is five years younger, Zach is a − 5, and in four years they are a + 4 and a − 1.

The signal words matter and are worth memorising: at most and no more than are ≤; at least and no less than are ≥; more than and fewer than are strict. The hardest question here runs the translation backwards — you are handed an equation and asked to invent a situation it describes, which is a much better test of whether you can read one than solving it would be.

Getting the most out of it

Write the check line every time

Under every solved equation, substitute your answer into the original and show both sides coming out equal. It costs one line, it catches nearly every arithmetic slip, and it is a habit that keeps paying for the rest of high school.

Solve one equation three ways

Take one of the bracket questions and do it by expanding, then by covering up, then by inverse operations. Same answer, three routes — and afterwards you will reach for the short one automatically instead of expanding everything out of habit.

Finish by asking what the letter stands for

Swims, driveways, bottles, dollars. The final line of a contextual question should name the quantity and its unit, and should use the number set that quantity actually lives in. That is the difference between solving an inequality and answering the question that was asked.

Common questions