Connect algebraic, graphical, and tabular representations of inequalities. | Linear inequalities in one or two variables | SAT Math

Move between inequality forms, graphs, and tables by matching the same solution set.

Core Idea

An inequality's solution set looks different in each form — an algebraic expression, a shaded region on a graph, or a set of rows in a table — but all three describe the same collection of values.

Understanding

The SAT may give you one representation and ask you to identify or work with another. The key is fluency in moving between them.

Algebraic → Graphical: Solve for $𝑦$ to get slope-intercept form, draw the boundary line, then shade. The inequality sign tells you which side.

Graphical → Algebraic: Read the slope and intercept from the boundary line, note whether the line is solid or dashed, and determine the shade direction to pick $<$ , $>$ , $\leq$ , or $\geq$ .

Tabular → Algebraic: Look at which $(𝑥, 𝑦)$ pairs satisfy the condition. The boundary between "yes" and "no" rows reveals the critical value. Check whether the boundary value itself is included (inclusive vs. strict).

Step by Step

Identify which representation you're given and which one you need.
If going to a graph: plot the boundary, choose solid or dashed, test a point to pick the shading side.
If going to algebra: read slope, intercept, and shade direction from the graph; choose the matching inequality symbol.
If working with a table: find where the inequality switches from true to false to locate the boundary value.

Misconceptions

Assuming a table that shows only integer values means the solution set is limited to integers — the solution set is continuous unless stated otherwise.
Reading the shading direction incorrectly when converting a graph back to an inequality — always verify with a test point.
Mixing up ≤ and < when the boundary value itself satisfies the inequality (solid line / closed circle means inclusive).

Question

Worked Example

A graph shows a solid line passing through $(0, 4)$ and $(2, 0)$ with the region below the line shaded. Which inequality represents the graph?

Select an answer to see the explanation