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4.6 – Linear Inequalities

Key Terms

  • xy-plane – A plane formed by a horizontal number line (the x-axis) and a vertical number line (the y-axis) that intersect at the zero point of each.
    • The xy-plane is also called the Cartesian coordinate system or coordinate plane.
  • Half-Plane – The graph of a linear inequality with two variables.
    • If the inequality sign is < or >, the line is dashed to show that it is not included in the solution.
    • If the inequality sign is \leq or \geq, the line is solid to show that it is included in the solution.

 


Review

  • < is less than
    • Ex: 2 < 6, two is less than six
  • > is greater than
    • Ex: 5 > 3, five is greater than three
  • \leq is less than or equal to (is at most…)
    • Ex: 4 \leq 4, four is less than or equal to four
    • Ex: 5 \leq 6, five is less than or equal to six
      • \leq 5, Megan is at most 5 feet tall (she could be 5 feet tall, but she could be shorter)
  • \geq is greater than or equal to (is at least…)
    • Ex: 7 \geq 7, seven is greater than or equal to seven
    • Ex: 8 \geq 7, eight is greater than or equal to seven
      • \geq 9, Zander is at least 9 years old (he may be 9 years old, but he could be older)
  • Remember: the sign is like a mouth.  The mouth eats more!

 

  • Plotting x and y
    • y is vertical (up and down) on an xy-plane
    • x is horizontal (left and right) on an xy-plane

 

  • Slope
    • Positive slopes rise from left to right
      • m = 4, m = 1/2, etc.
    • Negative slopes fall from left to right
      • m = -5, m= -1/8, etc.
    • Perpendicular slopes are negative reciprocals of each other
      • Slope -\frac{1}{4}\perp \frac{4}{1}
    • Steepness
      • A steeper slope has a greater number.  It doesn’t matter if it’s negative or positive.
      • m = -4 is steeper than m = 2 because 4 is bigger than 2.

 


Notes

  • When working with linear inequalities, you will have to shade above or below the line on the xy-plane.
    • This shading is called the half-plane.
    • The line will be either dashed or solid depending on the inequality.
      • Dashed: < or >
      • Solid: \leq or \geq
    • You will have to test out some points (using substitution) to see whether or not you shade above or below the line.
      • Alg1B 4.6 Test Points

 

  • There are actually two ways to determine how to shade the half-plane.
    • The first way: Replace, then Check, then Shade
      1. Replace the inequality symbol with an equals sign and graph the equation as a solid or dashed line.
      2. Check a point on either side of the line and note which one satisfies the inequality.
      3. Shade the half-plane that contains a point that satisfies the inequality.

 

  • Example: y > 2x + 3
    • Step 1: Draw the line on a graph and REPLACE the > with an equal sign: y = 2x + 3 to help you draw the line.
      • Draw a dashed line if the inequality symbol is > or <
      • In this case, the original inequality is >, so make it a dashed line (below)
    • Step 2: Choose a point on either side of that line to CHECK (test) the inequality
      • I chose (2.0) as it’s easy to test a point with a zero in it.
      • y > 2x + 3 would be  0 > 2(2) + 3
      • Simplify: 0 > 7 is false, so you do not shade near the chosen point.  Instead, SHADE on the OTHER side of the line.
      • Alg1B 4.6 Ex1 Graph No Shade ChoosePt
      • Alg1B 4.6 Ex1 Graph Shade Other Side

 

  • The second way is to look at the inequality after y.  Is it greater or less than the line?
    • If y > (slope & y-intercept), shade above the dashed line
    • y > xAlg1B 4.6 y Above Dashed
    • If y \geq (slope & y-intercept), shade above the solid line
    • y \geq xAlg1B 4.6 y Above Solid
    • If y < (slope & y-intercept), shade below the dashed line
    • y < xAlg1B 4.6 y Below Dashed
    • If y \leq (slope & y-intercept), shade below the solid line
    • y \leq xAlg1B 4.6 y Below Solid

 


Examples

  • Ex 1: Change this to slope-intercept form: y – 7 < 4(x – 3), draw the line, then shade the half-plane.
    • Step 1: distribute the 4
      • y – 7 < 4x – 12
    • Step 2: add 7 to both sides
      • y < 4x – 12 + 7
    • Step 3: simplify
      • y < 4x – 5
    • Step 4: Note the slope and y-intercept and plot them on a graph
      • Slope: 4 (so rise 4, run 1)
      • y-intercept: (0, -5)

Alg1B 4.6 Dashed Line

 

    • Step 5: Shade the half-plane
      • Since y < the line, shade below the line

Alg1B 4.6 Dashed Line Shade Below

 

  • Ex 2. Which of the following points satisfy the inequality: y\leq 3x-4
  • Choices:
    • A. (0, 0)
      • Test: 0\leq 3(0)-4 becomes 0\leq -4  No
    • B. (2, -2)
      • Test: -2\leq 3(2)-4 becomes -2\leq 2  Yes
    • C. (3, -6)
      • Test: -6\leq 3(3)-4 becomes -6\leq 5  Yes
    • D. (3, -8)
      • Test: -8\leq 3(3)-4 becomes -8\leq 5  Yes
    • E. (4, -5)
      • Test: -5\leq 3(4)-4 becomes -5\leq 8  Yes
    • F. (4, -8)
      • Test: -8\leq 3(4)-4 becomes -8\leq 8  Yes

 

  • Ex 3. A group of friends are hiking on trail Y when they reach an intersection with trail X.
    • Trail X is perpendicular to trail Y.
    • If trail Y has a slope of -\frac{1}{2}, which of the following statements are true?
      • We know that perpendicular slopes are negative reciprocals
        • Fact, trail Y: -\frac{1}{2}
        • So, trail X would be : \frac{2}{1}, or just 2.
        • Trail X is 2, which is a positive slope (it runs uphill).
        • Trail X is steeper than trail Y since 2 is a bigger number than 1/2
          • Steepness doesn’t matter if it’s negative or positive

 


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