Linear inequalities Revision Notes for AQA GCSE Further Maths

Linear inequalities

Understanding linear inequalities

Linear inequalities are mathematical expressions that show the relationship between two values using inequality symbols instead of equals signs. Unlike equations which have exact solutions, inequalities give us a range of possible values that satisfy the condition.

The main difference between equations and inequalities is that equations tell us exactly what a variable equals, while inequalities tell us what range of values a variable can take. For example, $x = 5$ has one solution, but $x > 5$ has infinitely many solutions.

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This fundamental difference between equations and inequalities is crucial to understand. While an equation like $x = 5$ has exactly one answer, an inequality like $x > 5$ represents all numbers greater than 5, which includes 5.1, 6, 100, 1000, and so on.

Inequality symbols and their meanings

When working with linear inequalities, you'll encounter these symbols:

$\<$ means "less than"
$>$ means "greater than"
$≤$ means "less than or equal to"
$≥$ means "greater than or equal to"

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The symbols $≤$ and $≥$ include the boundary value (the "equal to" part), while $\<$ and $>$ do not. This distinction becomes important when graphing solutions or checking boundary conditions.

Essential rules for solving linear inequalities

The process of solving linear inequalities is very similar to solving linear equations, with one crucial difference that you must always remember.

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The Golden Rule: When you multiply or divide both sides by a negative number, you must reverse the inequality sign.

This is the most important rule to remember and the most common source of mistakes in inequality problems.

Step-by-step method for solving linear inequalities

The systematic approach to solving linear inequalities follows the same algebraic principles as equations, but with careful attention to the direction of the inequality sign.

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Worked Example: Solving a Linear Inequality

Solve: $2y + 6 \< 5y + 12$

Method 1 (subtract the smaller variable term):

Subtract $2y$ from both sides: $6 \< 3y + 12$
Subtract $12$ from both sides: $-6 \< 3y$
Divide by $3$ : $-2 \< y$
Rewrite as: $y > -2$

Method 2 (subtract the larger variable term):

Subtract $5y$ from both sides: $-3y + 6 \< 12$
Subtract $6$ from both sides: $-3y \< 6$
Divide by $-3$ (remember to flip the sign!): $y > -2$

Both methods give the same answer, but notice how Method 2 required us to reverse the inequality sign when dividing by $-3$ .

Solving compound inequalities

Some inequalities contain a variable that appears between two values, creating what we call compound inequalities. These represent conditions where the variable must satisfy multiple constraints simultaneously.

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Worked Example: Compound Inequality

Solve: $5 \< 3x - 1 ≤ 17$

This type of inequality means the expression $3x - 1$ is greater than 5 AND less than or equal to 17.

Step-by-step solution:

Add $1$ throughout: $6 \< 3x ≤ 18$
Divide by $3$ : $2 \< x ≤ 6$

This means $x$ can be any value greater than 2 and less than or equal to 6.

Advanced inequality problems

More complex problems might involve finding ranges for expressions or dealing with squared terms. These problems require careful consideration of how operations affect the bounds of inequalities.

The table shows a sophisticated example involving finding inequality ranges for expressions like $x^2$ and $a - b$ when given constraints on the original variables. These problems require careful consideration of minimum and maximum values.

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Worked Example: Finding Expression Ranges

For finding the range of $a - b$ where $1 ≤ a ≤ 4$ and $-3 ≤ b ≤ 2$ :

The minimum value occurs when $a$ is smallest and $b$ is largest: $1 - 2 = -1$
The maximum value occurs when $a$ is largest and $b$ is smallest: $4 - (-3) = 7$
Therefore: $-1 ≤ a - b ≤ 7$

Common exam traps to avoid

Understanding where students commonly make mistakes can help you avoid these pitfalls in your own work.

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Major Pitfalls to Watch Out For:

Forgetting to reverse the inequality sign when multiplying or dividing by a negative number
Mixing up the direction of inequality signs when rearranging
Not considering both parts of compound inequalities properly
Incorrectly handling minimum and maximum values in range problems

These mistakes can completely change your final answer, so always double-check these critical steps!

Exam-style tips

Developing good problem-solving habits will help you tackle inequality questions with confidence and accuracy.

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Helpful Strategies:

Always check your final answer makes sense by substituting a test value
When dividing or multiplying by a negative, highlight this step to remind yourself to flip the sign
For compound inequalities, work on both inequalities simultaneously
Draw number lines to visualise your solution if it helps
Take extra care with boundary values in "less than or equal to" situations

Key takeaways

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Essential Points to Remember:

The golden rule: Flip the inequality sign when multiplying or dividing by a negative number
Inequalities show ranges: Unlike equations, inequalities give you a range of possible solutions
Check your work: Substitute a value from your solution back into the original inequality to verify
Compound inequalities: Treat them as two separate inequalities joined by "and"
Range problems: Find minimum and maximum values by considering extreme cases of the given constraints

Linear inequalities (AQA GCSE Further Maths): Revision Notes