Equations and Inequalities

The big idea is balance. Whatever you do to one side of the equation you must do to the other, otherwise the equals sign no longer holds. Think of it as a pair of scales: take $5$ off the left, you must take $5$ off the right.

You also work in reverse order. To peel away from $x$ you undo the operations one at a time, addition or subtraction first and then multiplication or division.

For an equation with the unknown on both sides, gather the letters on one side and the numbers on the other before you finish. For one with brackets, expand first or divide both sides by the bracket if the multiplier is a whole number. Always check your answer by substituting it back: in $3x + 5 = 20$, $3(5) + 5 = 20$ — perfect.

• An equation has one or more unknown letters and an equals sign.
• Do the same thing to both sides to keep the balance.
• Undo operations in reverse: add/subtract first, then multiply/divide.
• Always substitute your answer back to check.

For an equation with brackets, like $2(x + 4) = 18$, you have two clean options:

• Expand the bracket: $2x + 8 = 18$, then $2x = 10$, then $x = 5$.
• Divide by the multiplier first: $x + 4 = 9$, then $x = 5$.

Either path works — choose whichever is cleaner.

For fractional coefficients, clear the fraction by multiplying both sides by the denominator. So $\tfrac{x}{3} + 2 = 7$ becomes $x + 6 = 21$, giving $x = 15$. Or simply subtract first: $\tfrac{x}{3} = 5$, then $x = 15$. The path is up to you — pick the one with fewer slips.

• Move the smaller letter term first to avoid a negative coefficient.
• Expand brackets or divide by the multiplier — both are valid.
• Multiply through to clear a fraction when the denominator is a number.
• Always check by substituting your final answer back in.

There is one new rule, and it is the one to remember: if you multiply or divide both sides by a negative number, you must flip the inequality sign. So $-3x \ge 12$ becomes $x \le -4$ (divide by $-3$ and flip).

When you show an inequality on the number line, the picture tells the story:

• Open circle (hollow) for $<$ or $>$ — the endpoint itself is not included.
• Closed circle (filled) for $\le$ or $\ge$ — the endpoint is included.
• Draw a line or arrow towards the values that satisfy the inequality.

For a double inequality like $-1 < x \le 4$, both endpoints get a circle of their own (open at $-1$, closed at $4$) and the line connects them.

• Solve inequalities with the same balancing steps as equations.
• Flip the sign whenever you multiply or divide by a negative.
• Open circle = strict; closed circle = includes the endpoint.
• Double inequalities $a < x \le b$ need a circle at each end.

2. Substitution — rearrange one equation to make a letter the subject, then slot it into the other. Take $y = 2x + 1$ and $3x + y = 16$. Substitute the first into the second: $3x + (2x + 1) = 16$, so $5x = 15$, giving $x = 3$. Then $y = 2(3) + 1 = 7$.

Two quick tips. Match the coefficients first if a letter does not already line up. For $2x + y = 9$ and $x - y = 3$, the $y$ terms already cancel by adding — no rescaling needed. And always check both equations, not just the one you solved.

• Two equations in two unknowns can be solved together.
• Elimination — add or subtract to cancel one letter.
• Substitution — replace one letter using a rearranged equation.
• Substitute the pair $(x, y)$ back into both equations to check.

Word problems often hide inequalities, too. *"A taxi charges \3$plus$$2$per kilometre. How far can I travel for less than$$15$?"* Let the distance be$d$km. Then$3 + 2d < 15$, so$2d < 12$and$d < 6$km. Real situations sometimes need an *integer* answer — here you could travel up to$5$ km if the company only charges by whole kilometres. Always sense-check the answer against the story.

• Define the letter first — write down what it stands for.
• Build the equation phrase by phrase from the question.
• Solve carefully and re-read the question before you write the final line.
• Always sense-check that the answer fits the situation.

Step 2

1. Step 1

   Subtract $2x$ from both sides to gather the letters on the left.

   $3x - 4 = 11$

2. Step 2

   Add $4$ to both sides.

   $3x = 15$

3. Step 3

   Divide both sides by $3$.

   $x = 5$

4. Step 4

   Check: $5(5) - 4 = 21$ and $2(5) + 11 = 21$. ✓