Year 8AC9M8A02

Linear Equations and Inequalities

Graph linear relations on the Cartesian plane; solve linear equations and one-variable inequalities using graphical and algebraic techniques; verify solutions by substitution.

Solving an equation is undoing, one step at a time. An inequality works the same way, with one rule that catches everybody.

Builds onYr 7 · Solving Equations and Checking

Method

Do the same to both sides

The only rule. Everything else follows.

Undo in reverse order

Whatever was done last, undo first.

Flip the sign when you divide by a negative

-2x > 6 \Rightarrow x < -3

The single most common error in the topic.

Always verify by substitution

Ten seconds, and it catches every slip.

Worked example

e.g.

\text{Solve } \dfrac{3x - 5}{4} = 7 \text{ and then } -3x + 2 \leq 11

For the equation, undo the division first. $3x - 5 = 28$
Undo the subtraction. $3x = 33$
Undo the multiplication. $x = 11$
Verify by substituting back. $\frac{3(11)-5}{4} = 7 \quad\checkmark$
Now the inequality. Subtract 2 from both sides. $-3x \leq 9$
Divide by $-3$ — and flip the sign. $x \geq -3$

Practice

Solve, then check. For inequalities, ask whether you divided by a negative.

4x + 7 = 31

Answer

x = 6

\dfrac{x}{3} - 2 = 5

Answer

x = 21

2(x - 4) = 10

Answer

x = 9

5x - 3 < 12

Answer

x < 3

-4x \geq 20

Answer

x \leq -5

\text{Verify } x = 4 \text{ solves } 3x - 2 = 10

Answer

3(4)-2 = 10 \quad\checkmark