Equations and Inequalities
Equations and Inequalities
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Equations and Inequalities
Open with the idea that A-level questions rarely test just one skill: students must choose a method, carry it out accurately, and interpret the result set correctly.
Learning goal
I can solve and interpret a range of equations and inequalities using accurate algebra and clear exam-style working.
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What counts as a solution?
In this topic, a solution must do two things:
satisfy the equation or inequality
fit any restrictions from the expression or context
At A-level, marks are often lost by finding values and not checking whether they are valid.
Stress the difference between solving and validating. This becomes important for quadratics, rational equations and inequalities with excluded values.
Equations: finding exact values
An equation states that two expressions are equal.
Typical methods include:
rearranging linear equations
factorising quadratics
using the quadratic formula
solving by substitution after a change of variable
Keep the focus on method choice. Ask the student how they would recognise which method is efficient before any algebra begins.
Linear and quadratic equations
For quadratics, aim first to write the equation as
ax^2 + bx + c = 0
Then choose a method:
factorise if the factors are clear
complete the square if structure matters
use the quadratic formula if needed
Use standard notation carefully and verbally reinforce coefficient positions, especially the squared term.
Exam-style example
Solve
2x^2 - 7x + 3 = 0
Factorise:
(2x - 1)(x - 3) = 0
So x = 1/2 or x = 3
Model the zero-product rule explicitly. Highlight that 1/2 should be left exact, not turned into 0.5 unless the question asks for decimals.
When factorising is not obvious
For
3x^2 + 2x - 5 = 0
use x = (-b ± √(b^2 - 4ac)) / 2a
x = (-2 ± √64) / 6
x = 1 or x = -5/3
Emphasise the discriminant step b^2 - 4ac. It is a common checkpoint for arithmetic errors and helps students judge the number of real roots.
Exam technique for equations
write the equation in a standard form first
show each algebra step on a new line
keep exact values such as √5 or 1/2
substitute back if there is any doubt
give the final answer clearly as a set of values
This is especially useful for a resit student: encourage habits that earn method marks even if the final line goes wrong.
What should you usually do before solving a quadratic?
Set it equal to 0
Round the coefficients
Estimate the roots
Draw a graph first
Checks the standard-form habit from slides 4 to 7. Distractors reflect common weak exam habits: approximating too early or jumping to a graph.
Complete the exam-technique summary
For quadratics, first write the equation in standard form. If the factors are clear, factorise. If not, use the quadratic formula and keep answers in exact form such as 1/2 or √5. If needed, substitute back to check validity.
Echoes the wording of slides 4 to 7 and reinforces method choice plus exam presentation.
Inequalities: more than solving
An inequality describes a region of values, not usually a single answer.
Examples:
x > 3 means values greater than 3
x ≤ 5 means 5 is included
The sign tells you whether an endpoint is included.
Students often solve the related equation correctly but then state only the boundary values. Keep stressing interval thinking.
Solving linear inequalities
Solve as with an equation, but remember:
if you multiply or divide by a negative number, reverse the inequality sign
Example:
-3x + 4 > 10
-3x > 6
x < -2
Pause on why the sign reverses. A quick number-line explanation helps make this a rule with meaning, not memorisation.
Quadratic inequalities
To solve a quadratic inequality:
find the roots of the related equation
mark the roots on a number line
decide where the expression is positive or negative
Example structure:
(x - 2)(x + 1) > 0
Use sign analysis rather than guesswork. Students should understand that the sign changes at simple roots.
Worked quadratic inequality
Solve
(x - 2)(x + 1) > 0
Roots: x = -1 and x = 2
The product is positive outside the roots.
So x < -1 or x > 2
Point out that a strict inequality excludes the roots. If time allows, contrast with ≥ 0 on the same example.
Interval notation and presentation
Answers may be written as inequalities or intervals.
For the previous example:
x < -1 or x > 2
(-∞, -1) ∪ (2, ∞)
Use square brackets only when an endpoint is included.
Edexcel questions may accept either style depending on context, but the student must be consistent and accurate with inclusion.
When must the inequality sign reverse?
Add a negative
Multiply by zero
Divide by a negative
Factorise fully
Targets the key rule from slide 9. Students often remember to solve but forget the sign reversal.
Complete the inequality method
A quadratic inequality is solved by first finding the roots of the related equation. Then mark them on a number line and decide where the expression is positive or negative. For a strict inequality, the endpoints are excluded.
Reinforces slides 8 to 12: interval thinking, sign analysis and endpoint inclusion.
For (x - 2)(x + 1) > 0, where is the product positive?
-1 < x < 2
x < -1 or x > 2
x ≤ -1 or x ≥ 2
x = -1 or x = 2
Tests whether the student can move from roots to regions, not just identify boundary values.
Rational equations and restrictions
When x appears in a denominator, some values are not allowed.
Example restriction:
3 / (x - 2) is undefined when x = 2
So excluded values must be identified before or during solving.
This slide sets up the key idea behind extraneous solutions after multiplying through by an expression involving x.
Worked rational equation
Solve
1 / x + 1 / (x - 1) = 3
Restriction: x ≠ 0, 1
Multiply through by x(x - 1):
(x - 1) + x = 3x(x - 1)
Leave space for the student to continue mentally: 2x - 1 = 3x^2 - 3x, then 3x^2 - 5x + 1 = 0. The emphasis here is setting up the equation correctly and stating restrictions.
Finishing the rational equation
From
3x^2 - 5x + 1 = 0
x = (5 ± √13) / 6
Check against the restrictions x ≠ 0, 1.
Both values are valid, so both are solutions.
Stress that not every root of the cleared equation is automatically a solution of the original rational equation.
Rational inequalities
For rational inequalities, solve by sign analysis.
Example structure:
(x + 1) / (x - 3) < 0
Critical values come from:
numerator = 0
denominator = 0
Students often forget to include the denominator zero as a critical value even though it is not part of the solution set.
Worked rational inequality
Solve
(x + 1) / (x - 3) < 0
Critical values: x = -1, 3
The expression is negative for -1 < x < 3.
So the solution is -1 < x < 3
Reinforce that x = -1 is excluded here because the inequality is strict, and x = 3 is excluded because the expression is undefined.
Choosing a method quickly
A strong exam response begins by spotting the structure.
quadratic in x: factorise or use the formula
variable in a denominator: state restrictions
inequality: find critical values and test signs
awkward form: consider a substitution
This slide pulls together decision-making, which is often the hidden skill behind success on mixed exam questions.
Common errors to avoid
forgetting to reverse the sign after dividing by a negative
including values that make a denominator zero
giving boundary values instead of intervals
losing exact forms such as √13 or 5/3
factorising correctly but copying roots wrongly
Ask the student which of these feels most familiar from past papers. This can guide emphasis in live tutoring without turning the slide into an activity.
Why is x = 1 not allowed in 1 / (x - 1)?
It makes it negative
It makes it zero
It is not an integer
It makes denominator 0
Directly checks the restriction idea before students over-focus on algebraic manipulation.
Complete the rational-method summary
In a rational equation, values that make a denominator zero are excluded. After multiplying through, solve the new equation, then check each root against the original restrictions.
Anchors the restriction-checking process from slides 13 to 17.
Edexcel-style mixed question
Solve the inequality
x^2 - 5x + 6 ≥ 0
Factorise:
(x - 2)(x - 3) ≥ 0
So the solution is x ≤ 2 or x ≥ 3
Use this as a bridge between pure technique and exam presentation. Make the student justify why the middle region is excluded.
Edexcel-style mixed question
Solve
2 / (x + 1) = x
x ≠ -1
Multiply through by x + 1:
2 = x(x + 1)
x^2 + x - 2 = 0
x = 1 or x = -2
A neat example where both roots are valid. Good for contrasting with questions where one root must be rejected.
Edexcel-style mixed question
Solve
4x - 7 < 2x + 5
2x < 12
x < 6
Final answers should be stated clearly and on a single line.
Even easy-looking questions are a chance to insist on clean line-by-line working and a clearly isolated final result.
Summary
Across equations and inequalities, the same pattern helps:
identify the algebraic structure
choose an efficient method
solve accurately
check restrictions and endpoints
present the solution set clearly
End by tying method choice, accuracy and interpretation together. That is the central thread of the lesson and the strongest message for a resit student.
Which first step best fits 2 / (x + 1) = x?
State x ≠ -1
Reverse the sign
Use interval notation
Complete the square
Checks method selection from the mixed-question block: variable in a denominator means restrictions first.
Which error do you most need to watch for in exam questions?
Sign reversal
Restrictions
Intervals
Exact values
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What is one thing from today that you will remember?
Invite a brief reflection on the strongest takeaway, especially around method choice, restrictions, intervals or exact values.
Which part of this lesson felt hardest for you?
Use responses to diagnose where the student still needs rehearsal before the resit.
What helped you understand these methods best today?
Encourage metacognition: examples may include worked examples, sign analysis, the method checklist, or checking restrictions.
Glossary
equation - a statement that two expressions are equal
inequality - a statement comparing two expressions using signs such as <, >, ≤ or ≥
root - a value of x that makes an equation true
quadratic - an expression or equation involving x^2 as the highest power
discriminant - the quantity b^2 - 4ac in the quadratic formula
restriction - a value that is not allowed, such as one making a denominator zero
interval - a way of writing a set of values between or beyond endpoints
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