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5.5 – Two-Variable Systems of Inequalities

Key Terms

  • Axes – The reference lines used in a graph.
    • Axes is the plural of axis.
    • The axes of a coordinate graph are perpendicular, which means they intersect at 90° angles.
  • Axis – A reference line used in a graph.
    • The x-axis is horizontal, and the y-axis is vertical.
  • Boundary Line – A solid or dashed line on a graph that provides a border for the solution set.
  • Systems of Inequalities – Groups of inequalities that have the same variables and are used together to solve a problem.

 


Review

  • Greater Than:  >
    • Numbers of greater value
    • Numbers to the RIGHT on the number line
  • Less Than: <
    • Numbers of lesser value
    • Numbers to the LEFT on the number line
  • Greater Than or Equal to: ≥
    • Numbers of greater value OR equal value
    • “At Least”
  • Less Than or Equal to: ≤
    • Numbers of lesser value OR equal value
    • “At Most”

 

  • Graphing Linear Equations
    • Slope-Intercept Form: y = mx + b
      • m = slope (rise / run)
      • b = y-intercept (0, b)
      • Example: y = x + 2
        • Slope: 1
        • y-intercept (0, 2)

Alg1B 5.5 Shading Solid

 


Notes

  • Graphing Linear Inequalities
    • Graph lines in slope-intercept form: y = mx + b
      • Make the line dashed if y is > or < the line (mx + b)
      • Make the line solid if y ≥ or ≤ the line (mx + b)
    • Shade the half-plane that contains the solutions
      • If  y is GREATER than the line (mx + b): y > mx + b, shade ABOVE the line and make sure the line is DASHED
      • If   y is LESS than the line (mx + b): y < mx + b, shade BELOW the line and make sure the line is DASHED
      • If   y is GREATER THAN or EQUAL TO the line (mx + b): y ≥ mx + b, shade ABOVE the line and make sure the line is SOLID
      • If y is LESS THAN or EQUAL TO the line (mx + b): y ≤ mx + b, shade BELOW the line and make sure the line is SOLID

Alg1B 5.5 Shading on Graphs

 


  • Solution Set for a System of Two Inequalities
    • All the points in the area where the two graphed half-planes overlap.
    • To solve a system of inequalities graphically, you just need to graph each inequality and see which points are in the overlap of the graphs.
    • Example: The two lines are graphed and their solutions are shaded.
      • The pink is shaded above a dashed line
      • The blue is shaded below a solid line
      • The purple is the solution set for these two inequalities.
        • It includes ALL points on the solid line, but does NOT include any points on the dashed pink line.

Alg1B 5.5 Shading Overlap

 

      • Now, test a point in the overlap (purple) area to see if it works with BOTH inequalities

Alg1B 5.5 Shading Overlap Test


  • Parallel Lines
    • No Solution
      • Sometimes the graphs will never overlap
      • The lines will be parallel
      • The top line will have shading above it and the bottom line will have shading below it

Alg1B 5.5 Shading No Overlap

 

    • With Solutions
      • Parallel lines that have shading on top or on the bottom of BOTH lines will have overlap

Alg1B 5.5 Shading Parallel Overlap

 


Examples

  • Ex 1. Which of the following points are solutions to the system of inequalities shown below?
    • y > 5x -2
      y < 5x +3
    • Possibilities: (-5, 25), (4, 20), (5, 28), (4, 23)
      • Any points along a DASHED line are NOT part of the solution
      • Points MUST be inside the purple (overlap) section or along a SOLID line

Alg1B 5.5 Ex1Answer: only (4, 20)

  • Ex 2. Which of the following points are solutions to the system of inequalities shown below?
    • x+y \leq 4+3
    • y \textgreater 3
    • Possibilities: (3, 6), (2, -1), (1, 5), (1, 1), (-1, 8), (2, 8)
      • Points along the SOLID line ARE part of the solution
      • Points in the purple (overlap) section ARE part of the solution

Alg1B 5.5 Ex2Answers: (1, 5) and (-1, 8)

  • Ex 3. The graph below shows the solution to which system of inequalities?
    • Step 1: locate the y-intercepts
      • The dashed line crosses y at 0
      • The solid line crosses y at positive 3
    • Step 2: look at the rise over run (slope) for each line
      • The dashed line has a slope of \frac{1}{1}, so a slope of 1
      • The solid line has a zero slope (no rise, all run: horizontal line)
    • Step 3: write the equations for each line
      • Dashed: y = 1x + 0, which simplifies to y = x
      • Solid: y = 0x + 3, which simplifies to y = 3
    • Step 4: Look at the shaded area in relation to each line
      • Dashed: the shaded area is below it
      • Solid: the shaded area is above it
    • Step 5: Change the equal signs to inequality signs
      • Dashed: y = x becomes y < x (since all y-solutions are below it)
      • Solid: y = 3 becomes y \geq 3

Alg1B 5.5 Ex3

 


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