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Solving {x>−4,  x−2y≥−4}\left\{x > -4,\; x - 2y \geq -4\right\} by Graphing

Solve the system {x>−4x−2y≥−4\left\{\begin{array}{l} x > -4 \\ x - 2y \geq -4 \end{array}\right. by graphing.

Step 1 — Graph x>−4x > -4. The boundary line is x=−4x = -4, a vertical line passing through x=−4x = -4. Because the inequality uses >> (strict), draw a dashed line. Test (0,0)(0, 0): 0>−40 > -4 is true, so shade the side that contains the origin.

Step 2 — Graph x−2y≥−4x - 2y \geq -4 on the same grid. The boundary line is x−2y=−4x - 2y = -4, which has intercepts x=−4x = -4 and y=2y = 2. Because the inequality uses ≥\geq (non-strict), draw a solid line. Test (0,0)(0, 0): 0−2(0)≥−40 - 2(0) \geq -4 gives 0≥−40 \geq -4, which is true, so shade the side that contains the origin.

Step 3 — Identify the solution. The solution is the overlapping shaded region. The intersection point of the boundary lines is not included because one boundary line is dashed.

Step 4 — Verify with a test point. Choose a test point in the overlapping region and substitute it into both inequalities to confirm both are true.

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Updated 2026-04-28

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References

Tags

OpenStax

Intermediate Algebra @ OpenStax

Ch.4 Systems of Linear Equations - Intermediate Algebra @ OpenStax

Algebra

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