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All courses Algebra I · A-REI.12 8 of 76
Graph linear inequalities and systems of linear inequalities as half-plane solution regions.

Graph one linear inequality in slope-intercept form

Problem
For \(y\le~2x+1\), identify boundary equation, style, and shading direction.
Your answer
Show answer choicesHide answer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
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Hint

Graph the boundary line y = 2x + 1 first, then decide whether it should be solid or dashed.

Because the inequality uses <=, the boundary line is included, so it must be solid. Values less than or equal to y = 2x + 1 are below the line.

Solution walkthrough

01

Separate boundary from solution region

\[y\le~2x+1~->~\text{boundary}~y=2x+1\]

Replace the inequality symbol with equality to obtain the line that divides the plane into two half-planes.

02

Decode the boundary's graph features

\[y=2x+1;~\text{slope}=2;~y-\text{intercept}=1\]

The coefficient 2 means rise 2 for run 1, and the constant 1 gives the point (0,1). These features define the boundary line.

03

Choose solid boundary style

\[\le~\text{includes}~\text{equality}~->~\text{solid}\]

The phrase less than or equal to includes every point on y = 2x + 1, so the boundary cannot be dashed.

04

Determine the half-plane

\[y\le~2x+1~->~y-\text{values}~\text{below}~\text{or}~\text{on}~\text{the}~\text{line}\]

For any fixed x, solutions have y no greater than the boundary output. As a check, (0,0) gives 0 ≤ 1, so the side containing the origin is included.

05

State all graph specifications

\[\text{boundary}~=~y~=~2x~+~1;~\text{style}~=~\text{solid};~\text{shading}~=~\text{below}~\text{or}~\text{on}\]

The boundary equation, inclusion style, and shading direction answer the exact prompt and match choice A. The inspected answer SVG uses bounds x from -2 to 3 and y from -3 to 7 and shows this same region.

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Another way

  1. Test (0,0). Since 0 ≤ 2(0)+1 is true, shade the side containing the origin, including the boundary.

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Common mistake

Do not use a dashed boundary for an inclusive symbol or shade above because the slope is positive. Shading direction comes from comparing y with the line's output.

Coordinate graph showing the solid boundary line for y less than or equal to 2x plus 1, labeled to show the solution region is below the line.
Solid line y = 2x + 1; shade below the line.

Solution walkthrough video