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Systems of Inequalities Practice Problems

Decide line style and shading for each inequality, test points in every inequality, and read solution regions. The last problems turn a budget into a system of inequalities.

  1. 1.

    Is a solution of and ? Answer "yes" or "no".

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    ✓ and ✓.

  2. 2.

    Is a solution of and ? Answer "yes" or "no".

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    is false, so it fails the first inequality.

  3. 3.

    Is the boundary line of solid or dashed?

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    Strict inequality (), so the line is dashed: points on it are not solutions.

  4. 4.

    For , do you shade above or below the line?

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    Solved for with : shade above the line.

  5. 5.

    Is a solution of and ? Answer "yes" or "no".

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    ✓ and ✓.

  6. 6.

    Is a solution of and ? Answer "yes" or "no".

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    ✓ but is false. The point lies on the dashed boundary, which is excluded.

  7. 7.

    For , which way do you shade after solving for : above or below?

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    Divide by −1 and flip the sign: . Shade above.

  8. 8.

    Of the points , and , how many satisfy both and ?

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    : ✓ ✓. : ✗. : ✓ ✓. Two points.

  9. 9.

    The region , , is a triangle. Which vertex has the largest -coordinate? Give it as a pair.

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    The vertices are where the boundaries meet: , and . The largest is at .

  10. 10.

    Does the system and have any solutions? Answer "yes" or "no".

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    Parallel boundaries (slope 2); one region lies above the upper line and the other below the lower line, so they never overlap.

  11. 11.

    With $30, pens at $2 and notebooks at $5, and at least 3 notebooks: , . Can you buy 8 pens and 3 notebooks? Answer "yes" or "no".

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    , which is more than 30.

  12. 12.

    With the same budget system, what is the greatest number of pens you can buy if you buy exactly 4 notebooks?

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    gives , so at most 5 pens.
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Systems of Inequalities Practice Problems

These problems follow the systems of inequalities lesson. Use them as a graphing systems of inequalities worksheet: sketch each region before you answer.

Line style and shading

InequalityLineShade
dashedbelow
solidbelow
dashedabove
solidabove

Common mistakes to watch for

  • Shading one inequality only. The solution is where all shaded regions overlap.
  • Wrong boundary lines. Dashed for strict inequalities, solid for and .
  • Including dashed-boundary points. Points on a dashed boundary are not solutions.
  • Not testing a point. Check one point from the overlap in every inequality.

Frequently Asked Questions

When is the boundary solid?+

For ≤ and ≥, because points on the line are solutions. For < and > the line is dashed.

Which side do I shade?+

Solve for y: y > or y ≥ means above, y < or y ≤ means below. Or use a test point such as (0, 0).

How do I type my answer?+

Type yes or no, above or below, solid or dashed, a number, or a pair such as (5, 1), as the question asks.

More practice problems

Graphing Systems of Equations Practice Problems Systems of Equations Practice Problems Nonlinear Systems Practice Problems