Equations and Inequalities

The cure for these longer equations is one move per line, rewriting the whole equation each time. That single habit removes nearly every careless slip.

• Expand any brackets before anything else.
• Then gather all the letter terms onto one side.
• Solve the remaining equation with inverse operations.
• Write one move per line to stay accurate.

Left side: $3 \times (4 + 2) = 18$. Right side: $4 + 14 = 18$. They match, so $x = 4$ is right.

Always check using the original equation, not a line you have already rearranged — substituting into a rearranged line could hide the very mistake you are hunting for. Checking turns "I think it is right" into "I know it is right".

• Substitute your answer back into the equation.
• Use the original equation, never a rearranged line.
• Work out both sides — they should be equal.
• If they do not match, find and fix the slip.

So $4x - 3 < 17$ goes: add 3 to both sides for $4x < 20$; divide both sides by 4 for $x < 5$.

The one real difference is the answer. An equation usually has a single solution; an inequality has a range of them. $x < 5$ means every number below 5 works — 4, 3, 0, $-10$, even $4.9$. That range is what you show on a number line next.

• An inequality compares unequal amounts with $<$, $>$, $\le$, $\ge$.
• Solve it with the same balancing moves as an equation.
• Keep the inequality sign where the equals sign would be.
• The answer is a range of values, not a single number.

So $x < 5$ has an open circle at 5 with an arrow pointing left. By contrast, $x \ge 2$ has a filled circle at 2 with an arrow pointing right.

Read your solved inequality carefully: the sign chooses the circle, and which side the variable is on chooses the arrow direction.

• Show an inequality's range on a number line.
• Open circle for $<$ and $>$ — the end value is excluded.
• Filled circle for $\le$ and $\ge$ — the end value is included.
• The arrow points the way the solutions continue.

The solution is $x = 7,\ y = 3$ — and you can check it in both equations: $7 + 3 = 10$ and $7 - 3 = 4$. When the matching terms have the same sign, subtract instead of add.

• Simultaneous equations are two equations true at once.
• The solution satisfies both equations together.
• Add or subtract them to eliminate one letter.
• Substitute back to find the second letter, then check both.

If equation work feels hard later, it is usually the balance rule, expanding a bracket, or choosing add-or-subtract for a simultaneous pair that needs a quick refresh. There is no prize for rushing — accuracy first.

• Simultaneous equations become a major problem-solving tool.
• Straight-line graphs come straight from equations.
• Rearranging formulae reuses inverse operations.
• Equations and inequalities model real-life limits.

$x = 8$