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What is a Linear Inequality?

What is a linear inequality?

Summary

  1. Let's graph a linear equality (left) and a linear inequality (right) on x-y coordinate planes
  2. Notice that the inequality has a dashed boundary and a shaded half-plane!
  3. y=2x-2 is a linear equation
  4. y<2x-2 is a linear inequality
  5. '<' means 'less than'
  6. Ordered pairs outside the shaded area are not solutions
  7. Ordered pairs in the shaded area are solutions
  8. Since the inequality doesn't have a '≤' or '≥' symbol, boundary points are not solutions!

Notes

    1. To understand what a linear inequality is, let's compare linear inequalities with linear equations by graphing them
    1. The graph on the left in the diagram is a linear equation
    2. It's a solid line, and has an '=' sign when we write it down!
    3. In this example 'x' and 'y' stand for the x- and y- coordinates of points that satisfy the given equation
    1. The graph on the right in the diagram is a linear inequality
    2. Linear inequalities have inequality symbols like '<', '>', '≤', or '≥' when we write them down!
    3. In this example 'x' and 'y' stand for the x- and y- coordinates of points that satisfy the given inequality
    4. All of the points in the shaded region satisfy the given inequality
    1. The line that you draw for the inequality is called the boundary line
    2. The boundary line splits the coordinate plane into two half planes
    3. '<' and '>' stand for 'less than' and 'greater than'
    4. '≤' and '≥' stand for 'less than or equal to' and 'greater than or equal to'
    1. When you graph a line on a coordinate plane, you divide the plane into two pieces
    2. The two pieces are called 'half planes', and the dividing line is called the 'boundary'
    3. If your inequality has a '≤' or '≥' sign, then the solutions to the inequality come from the boundary and one of the half planes
    4. If your inequality has a '<' or '>' sign, then the solutions to the inequality come from one of the half planes but not from the boundary!
    1. Ordered pairs are usually written (x,y)
    2. When you have an inequality with two variables, the solutions to the inequality will be ordered pairs
    1. Let's see if the ordered pairs (3, 7) and (4,1) are solutions to the given inequality y<2x-2
    2. To check if (3,7) is a solution, plug in x=3 and y=7 and simplify
    3. (3,7) does not satisfy our inequality, so it's not part of the solution
    4. To check if (4,1) is a solution, plug in x=4 and y=1 and simplify
    5. (4,1) is part of our solution because it gives us a true statement when plugged in!
    1. In this case, the unshaded region falls to the left of the inequality
    2. (3, 7) falls into the unshaded region, so it is NOT a solution
    3. (4, 1) falls into the shaded region, so it IS a solution
    1. Let's see if the ordered pair (2,2) is a solution to the given inequality y<2x-2
    2. To check if (2,2) is a solution, plug in x=2 and y=2 and simplify
    3. Since the inequality doesn't have a '≤' or '≥' symbol, the point (2,2) is not part of the solution!
    1. A dashed line represents a '<' or '>', and any points on that line are not part of the solution
    2. A solid line represents a '≤' or '≥', and any points on that line are part of the solution