Absolute value
On the number line, absolute value is distance. This is the quickest intuition for equations and inequalities with absolute value.
Absolute value removes information about direction, but keeps information about length. On the number line, almost every absolute value question can be read as: how far away is the point?
The number x is a point on the number line. The value |x| tells how many units you must move from zero to reach that point.
That is why |3| = 3 and |-3| = 3. The points 3 and -3 are on different sides of zero, but they are equally far from it.
Example
|-6| = 6
The number -6 is left of zero, but its distance from zero is 6 units.
Interactive number line diagram
Choose the number x
Current absolute value
|-6|
The point -6 is left of zero, but its distance from zero has length 6.
Value: |-6| = 6
The point -6 is left of zero, but its distance from zero has length 6.
The expression |x - a| does not measure distance from zero, but distance from the number a. The point a becomes the new center of attention.
If |x - 3| = 5, then x is five units away from 3. You can move five units left or five units right, so x = -2 or x = 8.
Example
|x - 2| = 3
We want numbers 3 units away from 2, so the solutions lie symmetrically: x = -1 or x = 5.
Interactive number line diagram
Choose the value of a
Current equation
|x + 1| = 5
The point -1 is the center. The numbers -6 and 4 lie on both sides at distance 5.
Solutions: x = -6 or x = 4
The point -1 is the center. The numbers -6 and 4 lie on both sides at distance 5.
The distance between x and y is written as |x - y| or |y - x|. The order of subtraction does not change the result because distance has no sign.
This is useful when a problem asks how far two points are from each other, not which point is greater.
Example
|-4 - 3| = 7
The points -4 and 3 are on opposite sides of zero, so the distance between them has length 7.
Interactive number line diagram
Choose a pair of numbers
Distance
|7 - 2|
The points 2 and 7 are on the same side of zero. The segment between them has length 5.
Result: |7 - 2| = 5
The points 2 and 7 are on the same side of zero. The segment between them has length 5.
On the number line, absolute value turns a point into the length of a segment. The sign tells which side of zero the point is on, while absolute value tells how long the path to zero is.
In equations and inequalities, first mark the reference point, then ask whether x should be closer, farther away, or exactly at a given distance.
Example
|x| < 4
The condition means all points closer than 4 units to zero, so the inside of the interval from -4 to 4.
Interactive number line diagram
Choose a relation
Condition
|x| = 4
We want points exactly 4 units from zero.
Interpretation: x = -4 or x = 4
We want points exactly 4 units from zero.
An absolute value inequality describes points that are closer to or farther from a reference point. For a > 0 and r > 0, there are six basic patterns.
|x| < a
We want numbers whose distance from zero is less than a.
-a < x < a
Example
|x| < 3 gives -3 < x < 3
This is the inside of the interval between -a and a. The endpoints are excluded because < means strictly smaller distance.
Interactive number line diagram
Choose the value of a
Inequality
|x| < 2
Interval
-2 < x < 2
We want numbers closer than 2 units to zero.
|x| <= a
We want numbers whose distance from zero is at most a.
-a <= x <= a
Example
|x| <= 4 gives -4 <= x <= 4
This is the whole interval from -a to a including endpoints. The endpoints belong because the distance may equal a.
Interactive number line diagram
Choose the value of a
Inequality
|x| <= 2
Interval
-2 <= x <= 2
We want numbers at most 2 units away from zero.
|x| > a
We want numbers whose distance from zero is greater than a.
x < -a or x > a
Example
|x| > 2 gives x < -2 or x > 2
These are the two outside parts of the line: far left of zero or far right of zero. The middle is excluded.
Interactive number line diagram
Choose the value of a
Inequality
|x| > 2
Solution
x < -2 or x > 2
We want numbers farther than 2 units from zero.
|x| >= a
We want numbers whose distance from zero is at least a.
x <= -a or x >= a
Example
|x| >= 6 gives x <= -6 or x >= 6
Take everything outside the interval, including the boundary points.
Interactive number line diagram
Choose the value of a
Inequality
|x| >= 2
Solution
x <= -2 or x >= 2
We want numbers at least 2 units away from zero.
|x - a| < r
We want numbers closer than r units to the point a.
a - r < x < a + r
Example
|x - 2| < 5 gives -3 < x < 7
The point a is the center, and r acts like a radius to the left and right. The solution is the inside of that interval.
Interactive number line diagram
Choose a and r
Inequality
|x - 2| < 5
Interval
-3 < x < 7
The center is 2 and the radius is 5. The solution is the open interval from -3 to 7.
|x - a| > r
We want numbers farther than r units from the point a.
x < a - r or x > a + r
Example
|x + 1| > 3 gives x < -4 or x > 2
Remove the zone close to a. Two rays remain: left of a - r and right of a + r.
Interactive number line diagram
Choose a and r
Inequality
|x - 2| > 5
Solution
x < -3 or x > 7
The center is 2 and the radius is 5. Two rays remain outside -3 and 7.
Inequalities with < or <= mean points close to the center. Inequalities with > or >= mean points far from the center.
|x| = 5 has two solutions: -5 and 5. Only distance equal to 0 gives a single point.
The sign < gives open circles, while <= gives filled dots. The same idea applies to > and >=.
|x - a| is distance from a. You cannot split it into absolute value minus a number.
Absolute value is distance|x| measures the distance from point x to zero
Shifted center|x - a| measures the distance from point x to point a
Distance between two numbersThe distance between x and y is |x - y| = |y - x|
InequalitiesThe sign < leads to the inside interval, while > leads to two outside parts of the line
Next step
Now that you can read absolute value as distance on the number line, move on to equations. The same intuition helps you quickly find one, two, or no solutions.
Go to absolute value equations