Solve and interpret a linear inequation

For y > x + 1, does the point (1, 3) belong to the solution set?

Compare y > x + 1 with y ≥ x + 1. What changes?

Keep y = x + 1 fixed. Compare both shaded sides, then compare excluded and included boundaries. Decide which setting represents y ≤ x + 1.

Which two settings represent y ≤ x + 1?

From a boundary line to a half-plane

Replacing an inequality sign with equality gives its boundary line. The line separates the plane into two sides, called half-planes. Use a point away from the boundary to decide which side satisfies the original inequality

1. 1Boundary style Use a solid line for ≤ or ≥ and a dashed line for < or >
2. 2Test the original comparison A true result selects the side containing your test point. A false result selects the opposite side
3. 3The drawing has a window The shaded solution continues beyond the picture. The edges of the drawing are not extra boundaries

Which inequality matches this diagram?

For y ≤ x + 1, test (0, 0). Which conclusion is valid?

Why is (0, 0) not a useful side-test point for y > 2x?

For y > 2x, (0, 1) gives a true result. What should be shaded?

Which description matches y ≤ −2?

Which description matches x > 1?

Compare x ≥ 1 with x ≥ 3 by changing k. Then select the left side. Check the point (2, −4) using its x-coordinate

When k changes from 1 to 3 and the right side stays selected, what happens to (2, −4)?

Solve −2y ≥ 4x − 6 for y.

For −2y ≥ 4x − 6, which check detects choosing the wrong side?

Which point satisfies both x ≥ 0 and y ≤ 2 − x?

A stall can use at most $12 on supplies. Each small kit costs $2 and each large kit costs $3. Let x and y count small and large kits. Which model fits?

Under 2x + 3y ≤ 12 with non-negative integer counts, is (3, 2) feasible?

A stall buys small and large kits subject to 2x + 3y ≤ 12, where x and y are non-negative whole-number counts. Why does the continuous region for 2x + 3y ≤ 12 contain points that are not possible purchases?

Construct and justify your own region

On squared paper or a graphing tool, draw 2x + y < 4. State the boundary equation, find two boundary points, choose the correct line style and test one point on each side. Shade the satisfying side and explain why the region extends beyond your window

1. 1Self-check Boundary points (0,4) and (2,0) give equality. The line is dashed. (0,0) satisfies the inequality; (3,0) does not. Shade the side below the line, containing (0,0). The region continues beyond the drawing's window
2. 2Compare inclusion Change < to ≤. Keep the same side and make the boundary solid. Explain what was added