Math 9 · Linear equations and inequalities
Undo it, in reverse order
Solving an equation means finding the value that makes both sides equal. Every method in this unit is the same move: do the same thing to both sides until the variable is on its own. Inequalities work the same way, with one exception that costs more marks than anything else in the unit.
- 1. The words, first
- 2. Solving linear equations
- 3. Modelling with equations
- 4. Linear inequalities
- 5. What costs marks
The words, first
The idea: An equation says two things are equal; an inequality says which one is bigger. The same moves solve both, with one exception.
| Word | What it means |
|---|---|
| Equation | A statement that two expressions are equal: 3x + 7 = 22. |
| Solution | The value of the variable that makes the equation true. For 3x + 7 = 22 it is x = 5. |
| Inverse operation | The operation that undoes another: subtraction undoes addition, division undoes multiplication. |
| Verify | Substitute the solution into the original equation and check that both sides come out equal. |
| Distribute | Multiply the number outside a bracket by every term inside it: 3(x − 4) = 3x − 12. |
| Inequality | A statement using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). Its solution is every value that makes it true. |
| Open / closed dot | On a number line, an open dot means the endpoint is not included (< or >); a closed, filled dot means it is (≤ or ≥). |
Solving linear equations
The idea: Undo the operations in the reverse of the order they were done to x, and do exactly the same thing to both sides each time.
One step at a time. In 3x + 7 = 22, x was multiplied by 3 and then 7 was added. Undo in reverse: subtract 7, then divide by 3.
3x + 7 = 22 → 3x = 15 → x = 5 · verify: 3(5) + 7 = 22 ✓
Variables on both sides. 5x − 4 = 2x + 11. Subtract 2x from both sides so the variable is on one side only: 3x − 4 = 11, so 3x = 15 and x = 5.
Worked example with brackets and fractions. Solve ½(6x − 4) = ⅔(3x + 3).
- Distribute each fraction over its whole bracket: 3x − 2 = 2x + 2.
- Subtract 2x from both sides: x − 2 = 2.
- Add 2 to both sides: x = 4.
- Verify in the original: ½(24 − 4) = 10 and ⅔(12 + 3) = 10 ✓.
Division, either way round. Undo dividing by 4 by multiplying by 4. With x underneath, multiply both sides by x — it cannot be 0 there, because 12 ÷ 0 means nothing:
x/4 − 3 = 2 → x/4 = 5 → x = 20 · 12/x = 3 → 12 = 3x → x = 4
Modelling with equations
The idea: Say what the variable stands for, write one equation that says what the story says, solve it, and answer the question in words, with units.
Worked example. Gym A charges a $30 sign-up fee plus $12 a month. Gym B has no fee and charges $18 a month. After how many months do they cost the same?
- Let m be the number of months. Gym A costs 30 + 12m and Gym B costs 18m.
- Same cost means equal: 30 + 12m = 18m.
- Solve: subtract 12m from both sides, 30 = 6m, so m = 5.
- Check in the story: after 5 months, Gym A has cost 30 + 60 = $90 and Gym B 18 × 5 = $90 ✓.
- Answer the question: they cost the same after 5 months, and Gym A is cheaper after that.
A geometry one. A rectangle's length is 3 cm more than twice its width, and its perimeter is 54 cm. With width w, the length is 2w + 3:
2w + 2(2w + 3) = 54 → 6w + 6 = 54 → w = 8
So the rectangle is 8 cm by 19 cm. Check: 2(8) + 2(19) = 54 ✓. Stopping at w = 8 when the question asked for both measurements loses the mark.
Linear inequalities
The idea: Solve exactly as for an equation, with one exception: multiplying or dividing both sides by a negative number reverses the inequality sign.
Why. 2 < 5, but multiply both sides by −1 and −2 is the greater, because it is closer to zero: −2 > −5. The sign has to turn round to stay true. Adding and subtracting never do this.
Worked example 1. 2x − 1 < 5 → 2x < 6 → x < 3. Dividing by positive 2 changes nothing.
Worked example 2. −3x + 5 ≤ 11.
- Subtract 5 from both sides: −3x ≤ 6.
- Divide both sides by −3 and reverse the sign: x ≥ −2.
- Check the boundary: x = −2 gives 6 + 5 = 11, exactly equal ✓. Check the direction with a value from the answer, x = 0: 5 ≤ 11 ✓.
In context. Entry to a fair is $4 and each ride costs $3.50. With $20, how many rides can you take? 4 + 3.5r ≤ 20, so 3.5r ≤ 16 and r ≤ 4.57… Rides come whole, so the answer is at most 4 rides — not 4.57, and not 5.
What costs marks
The idea: One of these is the reversal rule. The rest are about doing the same thing to everything.
- Doing something to one side only, or to only one term on a side.
- Distributing to the first term only. 3(x − 4) is 3x − 12.
- Not verifying. Substituting takes seconds and catches nearly every slip.
- Forgetting to reverse the sign after multiplying or dividing by a negative — or reversing it after only adding or subtracting.
- Open and closed dots swapped.
- Answering 4.57 rides. The situation decides whether the answer has to be a whole number.