Graphical solution of linear inequalities in two variables

Substituting ( (0,0) ) gives (0>4), which is false, so the opposite side is taken. Exam tip: (>) does not include the boundary.

The origin gives (0\geq 12), which is false, so the side away from the origin is chosen. Exam tip: Always combine the region with the axes in the first quadrant.

The intersections of the boundary lines and their cuts on (y=0) give these three vertices. Exam tip: Solve boundary lines in pairs and verify each vertex.

The inequality (x+2y<8) is strict, so its boundary is not included. Exam tip: Use a dashed line for (<) and (>).

Since (2+2\cdot3=8) and (3\cdot2-3=3>1), the point is valid. Exam tip: A boundary point is excluded only for a strict inequality.

When removing the negative coefficient of (y), the inequality sign reverses. Exam tip: Never forget to reverse the sign when dividing by a negative number.

The axis directions give (x\geq 0), (y\geq 0), and being below the line gives (2x+y\leq 10). Exam tip: Convert words into axis inequalities first.

The same quantity (x+y) cannot be at most (4) and at least (7) simultaneously. Exam tip: Detect the empty gap between parallel boundary lines.

The intercepts are ((6,0)) and ((0,4)), so the area is (\frac{1}{2}\cdot6\cdot4=12). Exam tip: Use axis intercepts for the first-quadrant triangle.

The line (x+y=6) is not included, but the region remains within a triangle. Exam tip: A strict inequality changes closedness.

From the first two conditions, the maximum possible (x+y) is (5), so (x+y\geq 10) is impossible. Exam tip: Think about extreme limits before graphing.

At ((1,3)), both (2-3<1) and (1+3\geq 4) are true. Exam tip: Substitute options directly into the inequalities.

Substituting the origin gives (0\leq 6), so the side containing the origin is taken. Exam tip: ((0,0)) is the fastest test point when allowed.

The origin satisfies (x+y<5), so excluding it gives (x+y>5). Exam tip: If the boundary is excluded, choose only (<) or (>).

The two lines intersect at ((2,2)), and the axes give the limits ((3,0)) and ((0,3)). Exam tip: Verify all possible vertices with the inequalities.

Putting (y=0) gives (x=9), and putting (x=0) gives (y=3). Exam tip: Set one variable to zero to find intercepts.

Below means the value of (y) is less than the line value, and open boundary gives (<). Exam tip: In above-below questions, compare (y) directly.

The origin makes (0\geq 10) false, so the first-quadrant side away from the line is taken. Exam tip: With (\geq), the boundary line remains included.

The condition (x-y\leq 0) gives (y\geq x), and together with (x+y\leq 4) these form the triangle. Exam tip: Test every vertex in all inequalities.

At ((3,2)), (x+2y=7), but (x+2y<7) excludes the boundary. Exam tip: Remove equality points in strict inequalities.

The line is (x+2y=4), and the origin makes (0\geq 4) false. Exam tip: If the line is included, use only (\leq) or (\geq).

The two boundary lines meet at ((\frac{8}{3},\frac{8}{3})), where (x+y=\frac{16}{3}). Exam tip: A linear expression usually attains its extreme at a vertex.

Both conditions require (x+1\leq -x+5), which gives (x\leq 2). Exam tip: Compare the lower and upper boundary lines.

The origin satisfies (2x-5y\leq 10), so the opposite side is (2x-5y\geq 10). Exam tip: Keep equality for a solid boundary.

The condition (x-y\geq 0) gives (y\leq x), and its intersection with (3x+y=6) is ((\frac{3}{2},\frac{3}{2})). Exam tip: Reading the region is easier after rewriting inequalities.