Rearranging and Solving

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infoWhy this? Equations, inequalities, and rearranged formulae express relationships, constraints, and sets of possible values with precision. Solving and representing them develops logical reasoning, algebraic control, and the ability to model real situations.

scheduleWhy now? Building on balanced equation methods, brackets, fractions, and one-step rearrangement, this unit introduces unknowns on both sides, two-step changes of subject, and inequality solution sets. These skills support graphical reasoning and formula use across mathematics and science.

An equation states that two expressions are equal, so whatever you do to one side you must do to the other to keep the equality true.
To solve an equation you isolate the unknown by using inverse operations (for example, add/subtract are inverses and multiply/divide are inverses).
When an equation has unknowns on both sides, you can collect the unknown terms on one side by adding or subtracting the same term on both sides (for example, subtracting `3x` from both sides).
Like terms can be combined because they have the same variable part (for example, `7x-2x=5x` but `7x` and `2y` cannot be combined).
A one-step equation can be solved using a single inverse operation (for example, `x+5=12` gives `x=7`).
A two-step equation can be solved by reversing the operations in the reverse order (for example, `3x-4=11` gives `3x=15` then `x=5`).
A solution to an equation is a value of the unknown that makes the equation true when substituted back in.
An inequality compares two expressions using the symbols `<` (less than), `>` (greater than), `\le` (less than or equal to), and `\ge` (greater than or equal to).
A one-step inequality is solved using a single inverse operation, and the solution is usually a range of values rather than one number.
A two-step inequality is solved by reversing operations one at a time, just like equations, while keeping the inequality true.
If you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed (for example, from `<` to `>`).
If you add or subtract the same number on both sides of an inequality, the inequality sign stays the same.
On a number line, an open circle means the endpoint is not included (used with `<` or `>`), and a closed circle means the endpoint is included (used with `\le` or `\ge`).
On a number line, shading to the right shows values greater than the endpoint, and shading to the left shows values less than the endpoint.
A formula is an equation that shows a relationship between variables (for example, `A=lw` for area).
Changing the subject of a formula means rearranging it so a different variable is isolated on the left-hand side while keeping the equation balanced.
When changing the subject in two steps, you undo operations in the correct order, including handling brackets and division (for example, from `y=3x+5` to `x=\frac{y-5}{3}`).

rocket_launchYou must be able to

Solve equations with unknowns on both sides by collecting variable terms on one side and constants on the other, then isolating the variable using inverse operations.
Form one-step and two-step equations from written statements by translating words into operations and using a letter to represent the unknown consistently.
Solve one-step inequalities by applying the inverse operation to both sides and writing the solution as an inequality.
Solve two-step inequalities by undoing operations one at a time (often add/subtract first, then multiply/divide), keeping the inequality sign correct.
Reverse the inequality sign when multiplying or dividing both sides by a negative number, and justify this step in working.
Represent inequality solutions on a number line using correct endpoints (open/closed circles) and correct direction of shading.
Check solutions by substituting a test value into the original equation or inequality to confirm it makes the statement true.
Change the subject of a formula in two steps by applying inverse operations to both sides, keeping the target variable isolated and simplifying any fractions or bracketed expressions.

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