Writing Linear Inequalities from Graphs: Number Lines and Half-Planes

Algebra and Function Practice

Reading a graph to write its inequality is the reverse of graphing. You inspect the number line for endpoint style and direction, or the coordinate plane for the boundary line and shaded side. This lesson builds that translation skill with clear conventions, worked reasoning, and practice tasks you can check yourself.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Number Lines: Endpoints and Direction

A number line graph uses a closed dot for inclusive inequalities (≤ or ≥) and an open dot for strict inequalities (< or >). The shaded ray shows which values satisfy the inequality. Read the endpoint value first, then note whether the dot is open or closed, then note whether the ray extends left (less than) or right (greater than).

For example, a closed dot at 3 with shading to the right means x ≥ 3. An open dot at −2 with shading to the left means x < −2. If the shading covers the entire line, the inequality is all real numbers; if only a single point is marked, that is an equation, not an inequality.

1. Locate the endpoint on the number line and record its value.
2. Decide if the dot is open (strict) or closed (inclusive).
3. Observe the shaded direction: left means less than, right means greater than.
4. Combine endpoint, symbol, and variable to write the inequality.
5. Test one value from the shaded region and one from the unshaded region to confirm.

Coordinate Half-Planes: Boundary and Shading

A linear inequality in two variables graphs as a half-plane. The boundary line is solid for ≤ or ≥ and dashed for < or >. The shaded side contains the solutions. To write the inequality, first find the boundary line equation in slope-intercept form y = mx + b, then decide the symbol and direction from the shading.

A practical method is to pick a test point not on the boundary, such as (0,0) when possible. Substitute it into the boundary equation. If the test point lies in the shaded region, the inequality symbol should make the statement true for that point. If it lies outside, use the opposite symbol. For vertical boundaries, write x compared to a constant; for horizontal boundaries, write y compared to a constant.

1. Identify two points on the boundary line and compute slope m.
2. Use one point to find the y-intercept b, writing y = mx + b.
3. Check whether the boundary is solid (inclusive) or dashed (strict).
4. Choose a test point in the shaded region and substitute into y = mx + b.
5. Select the inequality symbol that makes the test point true, then verify with a point outside the shaded region.

Conventions and Valid Alternatives

For handwriting and diagram conventions in this lesson, define an open endpoint as a small circle with no fill and a closed endpoint as a filled circle. A dashed boundary line is drawn with evenly spaced short dashes; a solid boundary is continuous. These are conventions, not universal standards, and valid alternatives exist: some texts use brackets for inclusive endpoints, and some use different dash spacing. The rubric accepts any clearly labeled, consistent convention.

When writing inequalities, equivalent forms are acceptable. For example, x ≥ 3 can be written as 3 ≤ x, and y < 2x + 1 can be rearranged as 2x − y > −1. The rubric checks mathematical equivalence and correct endpoint or boundary treatment, not a single wording.

1. State your convention explicitly if it differs from open/closed circles or solid/dashed lines.
2. Keep the same convention throughout a single problem.
3. Accept equivalent algebraic forms that preserve the solution set.
4. Verify equivalence by testing at least one point in each region.

Worked example

A coordinate graph shows a solid boundary line passing through (0, 1) and (2, 5). The region below the line is shaded. Write the inequality.

Four practice exercises

Common mistakes

• Using a closed dot for a strict inequality such as x > 4. Match dot style to the symbol: open for < and >, closed for ≤ and ≥. Recheck the original graph's dot fill.
• Shading the wrong side of a boundary line because the test point was on the line. Choose a test point clearly off the boundary, such as (0,0) when the line does not pass through it, and substitute carefully.
• Writing y > mx + b when the shaded region is below the line. Use a test point in the shaded region. If it makes y < mx + b true, the symbol is < or ≤.
• Forgetting that a vertical boundary line has an equation like x = 2, not y = 2. Identify whether the boundary is vertical or horizontal before writing the equation; vertical lines compare x to a constant.

Self-check rubric

• Correct endpoint value or boundary equation identified.
• Correct symbol chosen from open/closed dot or dashed/solid line.
• Correct direction or shaded side represented by the inequality symbol.
• Equivalent algebraic forms accepted if the solution set is unchanged.
• Convention stated if it differs from open/closed circles or solid/dashed lines.
• Check method uses at least one point from each region.

Questions and answers