Math Problem Statement

The query submitted by the user involving solving a system of inequalities.

Solution

The image you provided outlines a step-by-step process for solving a system of inequalities. Let's go through the content in the image to help you understand and complete it:

Steps to Solve the System of Inequalities:

  1. Put the system of inequalities in general form:

    • Given: 5x+y35x + y \leq 3 x2x \geq 2
  2. Write one inequality in slope-intercept form (if applicable):

    • For 5x+y35x + y \leq 3, solve for yy: y5x+3y \leq -5x + 3
    • This form helps with graphing by showing the slope and y-intercept.
  3. Graph the inequalities on a coordinate plane:

    • For the inequality y5x+3y \leq -5x + 3:
      • Plot the line y=5x+3y = -5x + 3 as a solid line because the inequality includes "equal to" (\leq).
      • Shade below the line since yy is less than or equal to the line.
    • For the inequality x2x \geq 2:
      • Plot the vertical line x=2x = 2 as a solid line.
      • Shade to the right of the line since xx is greater than or equal to 2.
  4. Find the critical values in the shaded region:

    • Identify where the shaded regions overlap. This overlapping area is the solution to the system of inequalities.
  5. Label the area of interest on the graph:

    • Clearly indicate the feasible region that satisfies both inequalities.
  6. Check a point in the solution region:

    • Verify that the point satisfies both inequalities.

Questions Related to This Topic:

  1. How can you determine if a line should be dashed or solid when graphing inequalities?
  2. What does the slope-intercept form of a line reveal about the line?
  3. How can you verify the solution region of a system of inequalities?
  4. What is the significance of shading in graphing inequalities?
  5. How do you handle systems of inequalities with more than two inequalities?

Tip:

Always double-check your shaded regions when graphing inequalities to ensure that you have the correct overlap for your solution set.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Graphing

Formulas

Slope-intercept form y = mx + b

Theorems

Linear inequality theorem

Suitable Grade Level

Grades 8-10