M10 · Linear equations and inequalities | Edexcel GCSE Maths

Revise the key ideas

An equation's solution makes both sides equal. Keep an equation balanced by doing the same operation to each side; inverse operations help isolate the unknown.
Worked example: 5x − 7 = 18 gives 5x = 25 and x = 5. Check: 5 × 5 − 7 = 18.
Worked example: 4(x − 2) = 2x + 10 gives 4x − 8 = 2x + 10, then 2x = 18, so x = 9. Expand and collect before dividing.
For equations with fractions, multiply every term by a common denominator: x3 + 2 = 5 gives x + 6 = 15, so x = 9.
To form an equation, define the unknown. Three consecutive integers can be n, n + 1 and n + 2; if their sum is 24, 3n + 3 = 24, so they are 7, 8 and 9.
Graphically, solve f(x) = g(x) by reading the x-coordinates of intersections. The graph gives approximate solutions if its scale does not allow exact readings.

Solve a linear inequality like an equation, except multiplying or dividing by a negative number reverses the inequality sign. For −2x < 6, x > −3.
Worked example: 3x + 2 ≤ 14 gives 3x ≤ 12 and x ≤ 4. Equality is allowed, so 4 is included.
On a number line, use an open circle for < or > and a filled circle for ≤ or ≥. Shade or draw an arrow in the direction of the allowed values.
Number line for x less than or equal to four
Enlarge diagram: Number line for x less than or equal to four
For a double inequality, apply each operation to all three parts: 2 < 3x + 5 ≤ 11 gives −1 < x ≤ 2. The integer solutions are 0, 1 and 2.
The answer x < 4 describes infinitely many real numbers unless the question restricts x to integers. List integer solutions only when asked.

Set notation {x : x ≥ 2} means the set of x values satisfying x ≥ 2. Intersection means all conditions apply at once; union combines values allowed by either condition.
For an inequality in two variables, first draw its boundary line. Use a solid line if equality is allowed and a dashed line if it is not.
Test a point off the line to decide which side is allowed. For y > 2x + 1, (0, 0) fails, so the region is on the other side of the boundary.
For simultaneous inequalities, the solution is their overlapping region. State clearly whether the shaded region is the allowed region or the excluded region.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.