Inequalities, Intervals, and Absolute Value

Use inequalities and intervals to describe ordered sets, then use absolute value to express distance and tolerance. Track endpoints, unbounded ranges, negative scaling, and inside-versus-outside conditions.

An inequality compares values or describes a set of allowed values. Interval notation records that set compactly, while absolute value expresses distance without choosing a direction. Together, these ideas describe bounds, tolerances, thresholds, and constraints.

Inequalities Describe Ordered Sets

The statement

x<5

means that $x$ may be any real number smaller than $5$ . Unlike an equation with one solution, this inequality describes an entire set of values.

The four common comparison symbols are:

Symbol	Reading	Is equality included?
$<$	less than	no
$\le$	less than or equal to	yes
$>$	greater than	no
$\ge$	greater than or equal to	yes

Several inequalities can describe the same set. For example, $x\ge2$ and $2\le x$ both place $x$ at or to the right of $2$ on the number line.

Intervals Record Endpoints

An interval gives the lower and upper boundaries of a connected part of the number line. A square bracket includes a finite endpoint, while a parenthesis excludes it:

Inequality	Interval	Meaning
$0\le x\le1$	$[0,1]$	both endpoints included
$0<x<1$	$(0,1)$	both endpoints excluded
$0\le x<1$	$[0,1)$	$0$ included and $1$ excluded
$x\ge2$	$[2,\infty)$	every value from $2$ upward
$x<-3$	$(-\infty,-3)$	every value below $-3$

The symbols $\infty$ and $-\infty$ describe unbounded directions, not finite numbers that can be included. Their interval endpoints therefore always use parentheses.

Exercise: Write a mixed interval

Which interval represents $-2<x\le4$ ?

Choose the interval

(-2, 4]

[-2, 4)

[-2, 4]

Select one choice, then check.

HintRead one endpoint at a time

The symbol $<$ excludes $-2$ , while $\le$ includes $4$ .

SolutionMatch each endpoint

The lower bound is strict, so $-2$ receives a parenthesis. The upper bound includes equality, so $4$ receives a square bracket. Thus:

-2<x\le4 \quad\Longleftrightarrow\quad x\in(-2,4].

Reverse the Sign after Negative Scaling

Adding or subtracting the same quantity on both sides preserves an inequality. Multiplication or division by a positive number also preserves its direction. Multiplication or division by a negative number reverses the comparison.

Consider:

-2x<6.

Dividing both sides by $-2$ reverses $<$ to $>$ :

x>-3.

The reversal preserves the ordering. For example, $-4<-2$ , but dividing both values by $-2$ gives $2>1$ . A negative scale reflects the number line, so left and right exchange places.

If the sign of a multiplying expression is unknown, we cannot divide by it until we consider its positive, negative, and possibly zero cases separately.

Exercise: Reverse an inequality

Solve $-3x\ge12$ .

Choose the solution

x ≤ -4

x ≥ -4

x < -4

Select one choice, then check.

HintTrack the negative divisor

Divide both sides by $-3$ and reverse $\ge$ .

SolutionDivide and reverse the comparison

Dividing by $-3$ gives:

x\le\frac{12}{-3}=-4.

The endpoint remains included because the original comparison included equality. Substituting $x=-5$ confirms that $-3(-5)=15\ge12$ .

A compound inequality can be transformed across all three parts at once. For example:

-1<2x+3\le7

becomes

-4<2x\le4

after subtracting $3$ throughout, and then

-2<x\le2

after dividing throughout by the positive number $2$ .

Absolute Value Measures Distance

The absolute value of a real number is its distance from zero. It can be defined precisely as:

|x|= \begin{cases} x, & x\ge0,\\ -x, & x<0. \end{cases}

Thus $|3|=3$ and $|-3|=3$ . The result is always nonnegative. More generally,

|x-a|

measures the distance between $x$ and $a$ on the number line.

Exercise: Read absolute error

A prediction is $7.5$ and the target is $9$ . What is the absolute error $|7.5-9|$ ?

Your answer

Compute it first, then check your number.

HintDistance cannot be negative

Compute $7.5-9$ , then take its absolute value.

SolutionMeasure the distance

The signed difference is:

7.5-9=-1.5.

Absolute value keeps its magnitude:

|-1.5|=1.5.

The answer is a distance, so it is nonnegative.

Absolute-Value Bounds Describe Inside or Outside

For $r\ge0$ , the inequality

|x-a|\le r

means that $x$ lies within distance $r$ of the center $a$ . Equivalently:

a-r\le x\le a+r.

For example, $|x-10|\le2$ means:

8\le x\le12.

A lower bound on distance describes the outside instead. For $r\ge0$ :

|x-a|\ge r

means:

x\le a-r \quad\text{or}\quad x\ge a+r.

Strict comparisons produce strict endpoint inequalities in the corresponding forms.

Exercise: Convert a tolerance to an interval

Which interval is described by $|w-5|<0.5$ ?

Choose the interval

4.5 < w < 5.5

0.5 < w < 9.5

w < 4.5 or w > 5.5

Select one choice, then check.

HintUse a center and radius

The center is $5$ ; the allowed distance from it is less than $0.5$ .

SolutionFind both endpoints

Subtract and add the allowed distance:

5-0.5<w<5+0.5,

so:

4.5<w<5.5.

The endpoints are excluded because the original inequality is strict.

Exercise: Describe values outside a radius

Which condition is equivalent to $|x-3|>2$ ?

Choose the equivalent condition

1 < x < 5

x < 1 or x > 5

x > 5 only

Select one choice, then check.

HintFind the boundary points

Move $2$ units left and right from the center $3$ , then keep the values farther away than those points.

SolutionKeep both outer regions

The boundary points are $3-2=1$ and $3+2=5$ . A distance greater than $2$ places $x$ outside the interval between them:

|x-3|>2 \quad\Longleftrightarrow\quad x<1\ \text{or}\ x>5.

Both regions are required because distance has two directions from the center.

Name the Set Before Manipulating It

An inequality represents a set, not merely a symbolic arrangement. Before solving, identify whether the boundary is included, whether the set lies inside or outside a range, and whether any multiplication or division may reverse the order. After solving, test one value inside the proposed set and one outside it. This makes sign and endpoint errors easier to find.