Math Problem Statement

Is (1, 10) a solution to this system of inequalities?

y ≥ 8x + 2 y ≥ x + 10

Solution

To determine if (1,10)(1, 10) is a solution to the system of inequalities, we need to substitute x=1x = 1 and y=10y = 10 into each inequality and check if they hold true.

Step 1: Substitute (1,10)(1, 10) into each inequality

  1. First inequality: y8x+2y \geq 8x + 2

    • Substitute x=1x = 1 and y=10y = 10: 108(1)+210 \geq 8(1) + 2
    • Simplify the right side: 108+210 \geq 8 + 2 101010 \geq 10 This inequality holds because 10=1010 = 10.
  2. Second inequality: yx+10y \geq x + 10

    • Substitute x=1x = 1 and y=10y = 10: 101+1010 \geq 1 + 10
    • Simplify the right side: 101110 \geq 11 This inequality does not hold because 10<1110 < 11.

Conclusion

Since the point (1,10)(1, 10) does not satisfy the second inequality, it is not a solution to the system of inequalities.

Would you like further explanation on solving systems of inequalities?


  1. How can I determine if a point lies in the solution set of multiple inequalities?
  2. What strategies can I use to graph inequalities on a coordinate plane?
  3. How do I check solutions for inequalities with strict inequalities (e.g., >> or <<)?
  4. Can a point be a solution for one inequality but not the entire system?
  5. What is the difference between systems of inequalities and systems of equations?

Tip: Always test each inequality separately when working with systems of inequalities.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Systems of Inequalities

Formulas

y ≥ 8x + 2
y ≥ x + 10

Theorems

Substitution Method for Inequalities

Suitable Grade Level

Grades 9-11