Linear Inequalities and Number Lines

An equation says that two expressions are equal. An inequality compares their order instead. The four symbols in this lesson are

• $a<b$: $a$ is less than $b$;
• $a>b$: $a$ is greater than $b$;
• $a\leq b$: $a$ is less than or equal to $b$;
• $a\geq b$: $a$ is greater than or equal to $b$.

The extra line in $\leq$ or $\geq$ means that equality is allowed. Thus $6\leq6$ is true, whereas $6<6$ is false. Negative numbers retain their number-line order: $-5<-3$ because $-5$ lies farther left.

An inequality containing a variable usually has many solutions. For example, $x<4$ is satisfied by $3.9$, $0$ and $-20$, as well as infinitely many other values. By contrast, $x\leq4$ also includes the endpoint $4$.

Showing a solution set on a number line

A number line displays every permitted value, not just a few examples. Mark the boundary and extend the line or arrow towards the solutions:

• use an open circle when the boundary is excluded, as in $x<4$ or $x>-1$;
• use a filled circle when the boundary is included, as in $x\leq2$ or $x\geq0$.

The direction must agree with the inequality. Values less than a boundary lie to its left; values greater than it lie to its right.

A quick check prevents direction errors. For $x>-1$, the marked region should contain $0$ because $0>-1$ is true, but it should not contain $-2$.

Solving like an equation

To solve a linear inequality, isolate the variable while keeping the comparison equivalent. Adding or subtracting the same quantity on both sides preserves the order. Multiplying or dividing by the same positive number also preserves it.

For example:

Add $2$ to both sides and then divide by positive $3$:

The number line therefore has a filled circle at $4$ and extends to the right.

The variable may occur on both sides. For example:

Subtracting $3x$ and adding $4$ gives

These operations work for the same reason as they do in equations: they transform both sides in the same way without changing which values make the original statement true.

Why a negative multiplier reverses the sign

Multiplying or dividing both sides by a negative number reverses their order. For instance, $2<3$, but after multiplying both sides by $-1$ the correct comparison is $-2>-3$. Multiplication by $-1$ reflects the numbers across $0$ on the number line, so left and right exchange places.

Solve

Subtract $6$:

Now divide both sides by $-4$. Because the divisor is negative, reverse $\geq$ to $\leq$:

A check with $x=-4$ gives $6-4(-4)=22$, and $22\geq18$ is true. A value on the other side, such as $x=0$, makes the original statement false.

Only multiplication or division by a negative number forces this reversal. Merely adding or subtracting a negative number does not.

Compound inequalities

A compound inequality places two restrictions on the same variable. In

both comparisons must be true. Performing the same operation on all three parts keeps them connected. Add $5$ throughout and then divide throughout by positive $3$:

Its number line is the finite interval between $2$ and $5$, with an open circle at $2$ and a filled circle at $5$. The integer solutions are $3$, $4$ and $5$: $2$ is excluded, while $5$ is included.

You can alternatively split a compound inequality into two statements, solve both, and keep the values satisfying both results. Whichever method you use, the final number line must show their overlap.