Inequalities

Graphing on a Number Line

Represent a solved inequality on a number line. An open circle marks a strict endpoint (not included); a left-extending ray shows all values satisfying the inequality.

Example

Describe the boundary point and shading direction, then match them to the number-line picture.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

x<4x < 4
x < 41234567open circle

Step 2 — Boundary

Use an open circle at 4.

open circle at 4\hl{open} \ \mathrm{circle\ at}\ \hl{4}
x < 41234567open circle

Step 3 — Direction

Shade left from 4.

left from 4\hl{left} \ \mathrm{from}\ 4
x < 41234567open circle

Step 4 — Check a point

Check 3: 3 < 4, so it is on the shaded side.

3<4 shaded\hl{3} < 4 \ \mathrm{shaded}
x < 41234567open circle

Step 5 — Interval

Interval notation ends at 4.

x∈(−∞,4)x\in(-\infty, \hl{4} )
x < 41234567open circle
graph-on-number-line Open circle (o): endpoint NOT included (strict < or >). Closed circle: endpoint IS included (non-strict). The shaded ray extends toward all values in the solution set.