How are solutions found to a system of inequalities?

Solutions to a system of inequalities are found by identifying the set of ordered pairs that satisfy all the inequalities in the system simultaneously. This can be done graphically or algebraically.

Determining Solutions

To determine if an ordered pair is a solution:

• Substitution: We substitute the values of the variables into each inequality, as stated in the provided reference.
• Verification: If the ordered pair makes all inequalities true, then it is a solution to the system. If any of the inequalities are false, then it is not a solution.

Graphical Approach

Graphically, the solution to a system of inequalities is the region where the shaded areas of each inequality overlap. Here's how you'd find it:

1. Graph each inequality: This involves drawing the boundary line (solid if the inequality includes "equal to," dashed if it doesn't) and shading the region that satisfies the inequality.

2. Identify the overlapping region: The overlapping region represents the set of all points that satisfy all inequalities in the system. Any point within this region is a solution.

   • Example: Consider the following system:

     • y > x + 1
     • y < -x + 5

     The solution is the region where the shading for y > x + 1 and y < -x + 5 overlaps.

Algebraic Approach

While a graphical approach is helpful for visualization, algebraic methods are used to precisely identify solutions or verify if a specific ordered pair is a solution.

• Checking Ordered Pairs: Given a potential solution (an ordered pair), substitute the x and y values into each inequality.

  • Example: Is (1, 3) a solution to the system above?
    • For y > x + 1: 3 > 1 + 1 => 3 > 2 (True)
    • For y < -x + 5: 3 < -1 + 5 => 3 < 4 (True)
    • Since (1, 3) makes both inequalities true, it IS a solution.

• No Solution: It's possible for a system of inequalities to have no solution. Graphically, this would be represented by no overlapping shaded region.

Key Considerations

• Boundary Lines: Remember that points on a solid boundary line are included in the solution set, while points on a dashed boundary line are not.
• Test Points: You can use test points within each region to determine which side of the boundary line to shade. Choose a point not on the line, substitute its coordinates into the inequality, and see if it results in a true statement. If it does, shade that side of the line; if not, shade the other side.