Secondary 4 Equations and Inequalities Worksheet

All 10 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

   For each relation below, state whether it is an equation or an inequality, and name its type (first degree, second degree, square root, exponential, or greatest integer).

   1. $2^{x-1}=32$
   2. $\lfloor {\displaystyle \frac{x}{3}}\rfloor =4$
   3. $\sqrt{2x+5}\le 7$
   4. $3(x-2{)}^2-12=0$
   5. $5-4x>x+1$
2. Q2Representing Inequalities on a Cartesian Plane

   On the grid, represent the solution set of $y\le -\frac{2}{3}x+4$. State whether the boundary line is solid or dashed, and verify by testing the origin whether the shaded half-plane is the one containing $(0,0)$.

   A blank Cartesian grid for this question is on the printable PDF.

3. Q3Representing a Solution Set

   Write each solution set using interval notation.

   1. All real numbers greater than $-5$ and at most $2$.
   2. $\{x\in ℝ\mid x<-1\text{ or }x\ge 4\}$
   3. All real numbers except $3$.
4. Q4Solving Algebraic Inequalities

   Solve $5-2(x+3)>4x-7$ and express the solution set in interval notation.

5. Q5Solving Equations and Inequalities

   Solve ${\displaystyle \frac{2x-1}{3}}-{\displaystyle \frac{x+4}{2}}=1$.

6. Q6Solving a Greatest Integer Equation

   Solve $\lfloor 2x-1\rfloor =5$ and give the solution set in interval notation.

7. Q7Solving a Second Degree Equation or Inequality

   Solve $2x^2-7x-15=0$ by factoring.

8. Q8Solving a Square Root Equation or Inequality

   Solve $2\sqrt{x-1}+3=11$.

9. Q9Solving an Exponential Equation or Inequality

   Solve $5·2^{x+1}=160$.

10. Q10Synthesis — drawing on several sheets in this topic

    A market gardener builds a rectangular greenhouse whose length is $4$ m more than its width. The municipal permit requires a floor area of at most $96{\text{m}}^2$, and the supplier only delivers frames whose perimeter is at least $28$ m. Determine all possible widths and give the answer as an interval, in metres.

The 10 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 rule that costs the most marks

Everything you do to an equation, you may do to an inequality — with a single exception, and it is the exception that shows up in the marking. Multiplying or dividing both sides by a negative number reverses the sense of the inequality. Adding, subtracting, and multiplying by a positive number all leave it alone.

Interval notation: the bracket is the whole answer

Q3 is four seconds of work and a very common place to drop a mark, because the bracket carries the meaning. In Quebec notation the bracket turns outward at an end that is excluded:

Greatest integer: turn it into a double inequality

Q6 looks unfamiliar and is mechanical once you know the move. The floor of something equals k exactly when that something sits between k and the next integer:

Note the brackets: closed on the left, open on the right, always. Substitute your expression for u, solve the two inequalities together, and the answer comes out as an interval rather than a single number — which is the point of the question. A greatest integer equation almost never has one solution.

Square root: solve, then check — the check is part of the method

Q8 is a straightforward one: isolate the radical, square, solve. The habit it builds matters more than the answer.

Exponential: get the same base, then drop it

Q9 is the whole technique in one line. Once both sides are powers of the same base, the exponents must be equal, because an exponential function never takes the same value twice. Isolate the power first, then rewrite the number as a power: 32 is 2⁵, 81 is 3⁴, 243 is 3⁵.

One caution for the inequality version: comparing exponents preserves the sense only when the base is greater than 1. With a base between 0 and 1 the function is decreasing, and the inequality reverses — the same reflex as the negative-number rule, in different clothing.

Second degree: factor, then finish with the zero product

Q7 is a product-sum factoring: find two numbers whose product is a·c and whose sum is b, split the middle term into those two pieces, then group in pairs and take out the common bracket. The coefficient in front of x² is what makes the split necessary; with a coefficient of 1 you could have read the two numbers straight off.

Factoring is not the answer, though — it is the setup. The step that finishes the question is the zero product property: a product equals zero only when one of its factors does, so each bracket gives its own small first-degree equation. Say that out loud as you write it, because it is the only reason you are allowed to move from a factored form to a list of values, and it is also the reason this move works for = 0 and for nothing else. If the right-hand side is not zero, move everything across first; setting each factor equal to the number on the right is a genuine error, not a shortcut.

Two conditions at once

The last question puts two constraints on the same unknown. Solve each one separately, write each solution set as an interval, and then take the intersection — the values that satisfy both at once. Two things to watch: a constraint from the context (a width is positive; a number of sessions is a whole number) is as real as the algebraic one, and it is usually the constraint people forget.

One of those two constraints is where a second-degree inequality turns up, and there factoring really is only half the job. Bring everything to one side, factor, and then read the sign rather than the zeroes: written as (x − r₁)(x − r₂), a product of two factors is negative exactly between the zeroes and positive outside them, so the zeroes are the boundaries of the answer and not the answer itself. Sketching the parabola settles it in one line — where the curve sits below the axis is the solution to "< 0", and where it sits above is the solution to "> 0". Decide at the end whether the boundaries themselves belong, from whether the original sign was strict.

Preview all 4 pages

Click any page to open the full PDF.

Getting the most out of it

Write the restriction before you solve, not after

On the square root question, note "radicand ≥ 0" and "right-hand side ≥ 0" before you square anything. Ten seconds up front turns the check at the end from a guess into a confirmation.

Say the sense out loud when you divide

Every time you divide an inequality, say "by a positive, sense stays" or "by a negative, sense flips". It feels silly and it is the single highest-value habit on this sheet.

Do the classification question last, as a review

Q1 asks you to name each type rather than solve anything. Come back to it after the other nine and it stops being vocabulary and becomes a summary of what you just practised — which is how it tends to appear on an exam.

Common questions