Linear Inequalities and Number Lines

What an inequality tells you

An equation states that two expressions have the same value. An inequality compares their values. For example,

$x+3<8$

asks for every value of $x$ for which $x+3$ is less than $8$. There is usually a whole solution set, rather than just one answer.

The four main signs are:

• $<$ means “less than”;
• $>$ means “greater than”;
• $\leqslant$ means “less than or equal to”;
• $\geqslant$ means “greater than or equal to”.

The signs $<$ and $>$ give strict inequalities: equality is not allowed. The signs $\leqslant$ and $\geqslant$ are inclusive: the boundary value is allowed.

Solving a linear inequality in one variable

Most equation-solving operations also preserve an inequality. You can add or subtract the same quantity on both sides, expand brackets, collect like terms and divide by a positive coefficient.

For example,

$3x+4<19$ $3x<15$ $x<5$

This means that every number below $5$ is a solution. Substitution confirms the direction: $x=4$ gives $16<19$, which is true, whereas $x=6$ gives $22<19$, which is false.

Unknowns may occur on both sides. For example,

$5x-2\geqslant 2x+10$ $3x-2\geqslant 10$ $3x\geqslant 12$

Here $4$ is included because substituting it makes both original expressions equal.

Why a negative operation reverses the sign

Multiplying or dividing both sides by a negative number reverses their order. For instance, $2<5$, but multiplying both numbers by $-1$ gives $-2>-5$. Therefore the inequality sign must reverse whenever both sides are multiplied or divided by a negative number.

For example,

$7-2x>15$ $-2x>8$ $x<-4$

The last step divides both sides by $-2$, so $>$ becomes $<$. A quick check with $x=-5$ gives $7-2(-5)=17$, and is true.

You do not reverse the sign merely because a negative number appears somewhere. It reverses only when the operation applied to both sides is multiplication or division by a negative quantity.

Fractions, brackets and rearrangement

Use ordinary algebra to simplify before isolating the variable. For example,

$4(2x-1)\leqslant 3x+11$ $8x-4\leqslant 3x+11$

Fractions can be removed by multiplying every term by a common denominator. If that multiplier is positive, the sign stays the same. If it is negative, the sign reverses.

Occasionally the variable terms cancel. The result then describes either all values or no values. For example, $2(x+1)<2x+5$ simplifies to $2<5$, which is always true, so every real value of $x$ is a solution. By contrast, simplifies to , which is never true, so there are no solutions.

Representing a solution on a number line

A number line makes the entire solution set visible. First mark the boundary value, then show the correct side of it.

• Use an open circle for a strict inequality such as $x<3$ or $x>3$, because the boundary is excluded.

The circle records whether equality is allowed; the ray records which other values are solutions. For $x\geqslant-2$, for example, place a closed circle at $-2$ and draw to the right.

Double inequalities and finite intervals

A statement such as

$-1<x\leqslant4$

combines two conditions: $x>-1$ and $x\leqslant4$. Its number line has an open circle at $-1$, a closed circle at $4$, and a line segment between them. A value must satisfy both conditions.

When solving all three parts of a double inequality, perform the same operation throughout. For example,

$5<2x+1\leqslant13$ $4<2x\leqslant12$ $2<x\leqslant6$

Set notation

At Higher tier, a solution set may also be written in set notation. For example,

$\{x:x<5\}$

means “the set of values $x$ such that $x<5$”. Likewise, $\{x:-1<x\leqslant4\}$ describes all values between $$ and , excluding but including .

Linear inequalities in two variables

An inequality such as

$2x+y\leqslant7$

contains two variables. Its solutions are ordered pairs $(x,y)$, and there are usually infinitely many of them. To test $(2,3)$, substitute and :

$2(2)+3=7,$

so $(2,3)$ is a solution because equality is permitted. The pair $(3,2)$ is not a solution because $2(3)+2=8$, and $8$ is false. The order of the coordinates matters.

You can rearrange a two-variable inequality to make one variable the subject. For example,

$2x+y\leqslant7$ $y\leqslant7-2x.$

This form makes the condition on $y$ clear for any chosen value of $x$. If rearrangement requires division by a negative coefficient, reverse the sign. For instance,

$3x-2y>6$ $-2y>6-3x$ $y<\frac{3x-6}{2}.$

A graphical representation turns the equality case into a boundary line and shows a region of possible ordered pairs; the conventions and shading for those graphs are studied separately.