Equations and Inequalities

What you'll learn

How to solve linear inequalities safely, including when the inequality sign changes.
How to solve quadratic inequalities using roots and sign intervals.
How to combine two inequalities to find a shared solution set.
How simultaneous equations connect to intersections of lines, curves and circles.

1. The basic language

An equation says two expressions are equal. An inequality says one expression is bigger or smaller than another. Your answer is often not one value, but a whole range of values.

Definition

Solution set

A solution is a value that makes an equation or inequality true.
A solution set is the collection of all possible solutions.
Set notation writes this compactly. For example, ${x:x>3}$ means “the set of values of $x$ such that $x > 3$ ”.

2. Linear inequalities

A linear inequality is an inequality where the highest power of the variable is one, such as $5 x - 2 > 13$ .

You solve it almost like a linear equation: collect like terms, isolate the variable, and simplify.

Common Mistake

Dividing by a negative

If you multiply or divide both sides of an inequality by a negative number, the inequality sign reverses. For example, from $- 2 x > 6$ you get $x < - 3$ , not $x > - 3$ .

Multiplying or dividing an inequality by a negative reverses the order of the values on the number line.

Example

Solving a linear inequality

Solve $20 - x < 4 x + 5$ .

Subtract 5 from both sides.

$15 - x < 4 x$
Add $x$ to both sides to collect the variable terms.

$15 < 5 x$
Divide by positive 5, so the inequality sign stays the same.

$3 < x$
Write the answer in the usual order.

$x > 3$

3. Quadratic inequalities

A quadratic inequality contains a squared term, such as $x^2+6x>16$ .

The key skill is to compare the quadratic with zero. First solve the matching quadratic equation. The roots are the boundary values where the expression changes sign.

Key Idea

Roots split the number line

For a factorised quadratic, the roots split the number line into intervals. Test the sign on each interval, or use the shape of the parabola.

For a positive $x^2$ coefficient, the parabola opens upwards. It is positive outside the roots and negative between the roots.

An upward-opening quadratic is above the x-axis outside its roots and below the x-axis between them.

Quadratic sign diagram showing positive outside the roots and negative between them

Example

Solving a quadratic inequality

Solve $x^2+6x>16$ , giving your answer in set notation.

Move everything to one side so you compare with zero.

$x^2+6x-16>0$
Factorise the quadratic.

$(x + 8) (x - 2) > 0$
Find the boundary values by solving $(x + 8) (x - 2) = 0$ .

$\quad \text{or} \quad x=2$
Since the coefficient of $x^2$ is positive, the quadratic is greater than zero outside the roots.

$\text{ or } x>2}$

Strict and non-strict inequalities

A strict inequality uses $<$ or $>$ , so the boundary values are not included.

A non-strict inequality uses $≤\le$ or $≥\ge$ , so the boundary values are included if they satisfy the inequality.

Example

Including the endpoints

Solve $x(x+2)≤15x(x+2)\le15$ .

For a non-strict quadratic inequality, the endpoints are included when the solution is between the roots.

Expand and move everything to one side.

$x2+2x−15≤0x^2+2x-15\le0$
Factorise.

$(x+5)(x−3)≤0(x+5)(x-3)\le0$
The boundary values are $x = - 5$ and $x = 3$ .
The quadratic opens upwards, so it is less than or equal to zero between the roots, including the endpoints.

$−5≤x≤3-5\le x\le3$

Tip

Quick sign check

If you are unsure which interval to choose, test one easy value. For $(x + 5) (x - 3)$ , testing $x = 0$ gives $5×(−3)5\times(-3)$ , which is negative, so the middle interval works for “less than zero”.

4. Combining inequalities

Sometimes you are asked for values that satisfy two inequalities at the same time. Solve each one separately, then take the overlap.

Combining inequalities means keeping only the part of the number line common to both solution sets.

Example

Finding the shared solution set

Find the values of $x$ which satisfy both $x^2+3x-18<0$ and $15+3x≤17+x15+3x\le17+x$ .

Solve the quadratic inequality first.

$x^2+3x-18<0$
Factorise and identify the roots.

$(x + 6) (x - 3) < 0$
The quadratic is negative between the roots.

$- 6 < x < 3$
Now solve the linear inequality.

$15+3x≤17+x15+3x\le17+x$
Rearrange.

$2x≤22x\le2$
Divide by positive 2.

$x≤1x\le1$
Combine the two results by taking the overlap.

${x:−6<x≤1}{x:-6<x\le1}$

5. Simultaneous equations and intersections

Simultaneous equations are equations that must be true at the same time. When one equation is a line and the other is a curve, the solutions are the coordinates of their intersection points.

The solutions of simultaneous equations are the coordinates where the graphs meet.

Usually, you substitute the line equation into the curve equation. This often produces a quadratic equation.

A line intersecting a parabola, showing that simultaneous equations give intersection points

Example

Line and curve intersection

Find the coordinates where the curve $C$ with equation $y=x^2-4x+6$ meets the line $L$ with equation $x + y = 6$ .

Rearrange the line to make $y$ the subject.

$y = 6 - x$
Substitute this into the curve equation.

$6-x=x^2-4x+6$
Simplify to form a quadratic equation.

$x^2-3x=0$
Factorise and solve.

$x (x - 3) = 0$
So the possible $x$ -values are:

$\quad \text{or} \quad x=3$
Substitute each value into $y = 6 - x$ .

$\quad \text{and} \quad (3,3)$

6. When a line does not meet a curve

The discriminant of a quadratic $ax^2+bx+c=0$ is $b^2-4ac$ . It tells you how many real roots the quadratic has.

The discriminant determines whether a line and a quadratic curve meet twice, touch once, or do not meet.

If $b^2-4ac>0$ , there are two real roots.
If $b^2-4ac=0$ , there is one repeated root.
If $b^2-4ac<0$ , there are no real roots.

So if a line does not intersect a curve, the quadratic formed by substitution has no real solutions.

Example

Using the discriminant with a parameter

The curve $C$ has equation $y=px^2-2px-2p$ , where $p$ is a constant. The line $L$ has equation $y = x - 3$ . Find the values of $p$ for which $C$ and $L$ do not intersect.

Set the two expressions for $y$ equal.

$px^2-2px-2p=x-3$
Rearrange into quadratic form.

$px^2-(2p+1)x+(3-2p)=0$
For no intersection, the discriminant must be less than zero.

$2p+1))^2-4p(3-2p)<0$
Expand and simplify.

$12p^2-8p+1<0$
Factorise.

$(6 p - 1) (2 p - 1) < 0$
The quadratic in $p$ is negative between its roots.

$16<p<12\frac{1}{6}<p<\frac{1}{2}$

Exam technique

In the exam

Move all terms to one side before solving a quadratic inequality.
Always decide whether endpoints are included: use open intervals for $<$ and $>$ , closed endpoints for $≤\le$ and $≥\ge$ .
For intersections, substitute the line into the curve, solve for one variable, then substitute back for the other coordinate.

Self review

Check yourself

When solving an inequality, what operation makes the inequality sign reverse?
Why does a quadratic inequality usually have an interval answer rather than one value?
If a line and curve do not intersect, what must be true about the discriminant?

What you'll learn

How to solve linear inequalities safely, including when the inequality sign changes.
How to solve quadratic inequalities using roots and sign intervals.
How to combine two inequalities to find a shared solution set.
How simultaneous equations connect to intersections of lines, curves and circles.

1. The basic language

An equation says two expressions are equal. An inequality says one expression is bigger or smaller than another. Your answer is often not one value, but a whole range of values.

Definition

Solution set

A solution is a value that makes an equation or inequality true.
A solution set is the collection of all possible solutions.
Set notation writes this compactly. For example, ${x:x>3}$ means “the set of values of $x$ such that $x > 3$ ”.

2. Linear inequalities

A linear inequality is an inequality where the highest power of the variable is one, such as $5 x - 2 > 13$ .

You solve it almost like a linear equation: collect like terms, isolate the variable, and simplify.

Common Mistake

Dividing by a negative

If you multiply or divide both sides of an inequality by a negative number, the inequality sign reverses. For example, from $- 2 x > 6$ you get $x < - 3$ , not $x > - 3$ .

Multiplying or dividing an inequality by a negative reverses the order of the values on the number line.

Example

Solving a linear inequality

Solve $20 - x < 4 x + 5$ .

Subtract 5 from both sides.

$15 - x < 4 x$
Add $x$ to both sides to collect the variable terms.

$15 < 5 x$
Divide by positive 5, so the inequality sign stays the same.

$3 < x$
Write the answer in the usual order.

$x > 3$

3. Quadratic inequalities

A quadratic inequality contains a squared term, such as $x^2+6x>16$ .

The key skill is to compare the quadratic with zero. First solve the matching quadratic equation. The roots are the boundary values where the expression changes sign.

Key Idea

Roots split the number line

For a factorised quadratic, the roots split the number line into intervals. Test the sign on each interval, or use the shape of the parabola.

For a positive $x^2$ coefficient, the parabola opens upwards. It is positive outside the roots and negative between the roots.

An upward-opening quadratic is above the x-axis outside its roots and below the x-axis between them.

Quadratic sign diagram showing positive outside the roots and negative between them

Example

Solving a quadratic inequality

Solve $x^2+6x>16$ , giving your answer in set notation.

Move everything to one side so you compare with zero.

$x^2+6x-16>0$
Factorise the quadratic.

$(x + 8) (x - 2) > 0$
Find the boundary values by solving $(x + 8) (x - 2) = 0$ .

$\quad \text{or} \quad x=2$
Since the coefficient of $x^2$ is positive, the quadratic is greater than zero outside the roots.

$\text{ or } x>2}$

Strict and non-strict inequalities

A strict inequality uses $<$ or $>$ , so the boundary values are not included.

A non-strict inequality uses $≤\le$ or $≥\ge$ , so the boundary values are included if they satisfy the inequality.

Example

Including the endpoints

Solve $x(x+2)≤15x(x+2)\le15$ .

For a non-strict quadratic inequality, the endpoints are included when the solution is between the roots.

Expand and move everything to one side.

$x2+2x−15≤0x^2+2x-15\le0$
Factorise.

$(x+5)(x−3)≤0(x+5)(x-3)\le0$
The boundary values are $x = - 5$ and $x = 3$ .
The quadratic opens upwards, so it is less than or equal to zero between the roots, including the endpoints.

$−5≤x≤3-5\le x\le3$

Tip

Quick sign check

4. Combining inequalities

Sometimes you are asked for values that satisfy two inequalities at the same time. Solve each one separately, then take the overlap.

Combining inequalities means keeping only the part of the number line common to both solution sets.

Example

Finding the shared solution set

Find the values of $x$ which satisfy both $x^2+3x-18<0$ and $15+3x≤17+x15+3x\le17+x$ .

Solve the quadratic inequality first.

$x^2+3x-18<0$
Factorise and identify the roots.

$(x + 6) (x - 3) < 0$
The quadratic is negative between the roots.

$- 6 < x < 3$
Now solve the linear inequality.

$15+3x≤17+x15+3x\le17+x$
Rearrange.

$2x≤22x\le2$
Divide by positive 2.

$x≤1x\le1$
Combine the two results by taking the overlap.

${x:−6<x≤1}{x:-6<x\le1}$

5. Simultaneous equations and intersections

Simultaneous equations are equations that must be true at the same time. When one equation is a line and the other is a curve, the solutions are the coordinates of their intersection points.

The solutions of simultaneous equations are the coordinates where the graphs meet.

Usually, you substitute the line equation into the curve equation. This often produces a quadratic equation.

A line intersecting a parabola, showing that simultaneous equations give intersection points

Example

Line and curve intersection

Find the coordinates where the curve $C$ with equation $y=x^2-4x+6$ meets the line $L$ with equation $x + y = 6$ .

Rearrange the line to make $y$ the subject.

$y = 6 - x$
Substitute this into the curve equation.

$6-x=x^2-4x+6$
Simplify to form a quadratic equation.

$x^2-3x=0$
Factorise and solve.

$x (x - 3) = 0$
So the possible $x$ -values are:

$\quad \text{or} \quad x=3$
Substitute each value into $y = 6 - x$ .

$\quad \text{and} \quad (3,3)$

6. When a line does not meet a curve

The discriminant of a quadratic $ax^2+bx+c=0$ is $b^2-4ac$ . It tells you how many real roots the quadratic has.

The discriminant determines whether a line and a quadratic curve meet twice, touch once, or do not meet.

If $b^2-4ac>0$ , there are two real roots.
If $b^2-4ac=0$ , there is one repeated root.
If $b^2-4ac<0$ , there are no real roots.

So if a line does not intersect a curve, the quadratic formed by substitution has no real solutions.

Example

Using the discriminant with a parameter

The curve $C$ has equation $y=px^2-2px-2p$ , where $p$ is a constant. The line $L$ has equation $y = x - 3$ . Find the values of $p$ for which $C$ and $L$ do not intersect.

Set the two expressions for $y$ equal.

$px^2-2px-2p=x-3$
Rearrange into quadratic form.

$px^2-(2p+1)x+(3-2p)=0$
For no intersection, the discriminant must be less than zero.

$2p+1))^2-4p(3-2p)<0$
Expand and simplify.

$12p^2-8p+1<0$
Factorise.

$(6 p - 1) (2 p - 1) < 0$
The quadratic in $p$ is negative between its roots.

$16<p<12\frac{1}{6}<p<\frac{1}{2}$

Exam technique

In the exam

Move all terms to one side before solving a quadratic inequality.
Always decide whether endpoints are included: use open intervals for $<$ and $>$ , closed endpoints for $≤\le$ and $≥\ge$ .
For intersections, substitute the line into the curve, solve for one variable, then substitute back for the other coordinate.

Self review

Check yourself

When solving an inequality, what operation makes the inequality sign reverse?
Why does a quadratic inequality usually have an interval answer rather than one value?
If a line and curve do not intersect, what must be true about the discriminant?

What you'll learn

1. The basic language

2. Linear inequalities

3. Quadratic inequalities

Strict and non-strict inequalities

4. Combining inequalities

5. Simultaneous equations and intersections

6. When a line does not meet a curve

Recap questions

Review Equations and Inequalities by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Equations and Inequalities

What you'll learn

1. The basic language

2. Linear inequalities

3. Quadratic inequalities

Strict and non-strict inequalities

4. Combining inequalities

5. Simultaneous equations and intersections

6. When a line does not meet a curve

Recap questions

Review Equations and Inequalities by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Equations and Inequalities

What you'll learn

1. The basic language

2. Linear inequalities

3. Quadratic inequalities

Strict and non-strict inequalities

4. Combining inequalities

5. Simultaneous equations and intersections

6. When a line does not meet a curve

Recap questions

Review Equations and Inequalities by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Equations and Inequalities Revision Guide

Equations and Inequalities

What you'll learn

1. The basic language

2. Linear inequalities

3. Quadratic inequalities

Strict and non-strict inequalities

4. Combining inequalities

5. Simultaneous equations and intersections

6. When a line does not meet a curve

Recap questions

Review Equations and Inequalities by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Equations and Inequalities Revision Guide