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College Algebra MT 150

Day #3: Section 3.5 – Linear Inequalities in Two Variables

Today, we want to accomplish 3 things:

  1. We want to be able to graph linear inequalities on a number line.
  2. We want to be able to graph linear inequalities on graph paper.
  3. We want to be able to graph an absolute value inequality.

Let’s review a little bit about inequalities:

  1. When working with inequalities (<, >, ), you should get the ‘variable’ on the LEFT and the numbers on the RIGHT.
  2. When you are working with an equation (=), you can either have the variable on the RIGHT or LEFT side of the = sign, but NOT with inequalities.

Let’s review a little bit about INTERVAL NOTATION, before we begin graphing linear inequalities on a number line.

  1. We use interval notation to describe the ‘solution set’ when you are working with a number line.
  2. Use brackets [ ], when inequality has either.
  3. Use parentheses ( ) when inequality has either < or >.
  4. When shading, if you shade a number all the way to the END of the number line, use or - in the interval notation, depending on the direction of the shading.
  5. When you use either positive or negative infinity, we ALWAYS use a parenthesis.

EXAMPLES:

Instructions: Solve the following inequalities with one variable. Describe the solution set using interval notation and by graphing on a number line.

  1. 4 + 3t t – 2 2. 5y – 24 < -9.6 + 2y
  1. -2 (3 – x) < -2x4.
  1. -4 < 3x – 7 6. -12 < -3 (5 + y) 9
  1. 4 < 98. 4 < 9

Let’s now work towards graphing inequalities graphically, by reviewing how to graph an inequality versus an equation and distinguishing between ‘AND’ & ‘OR’.

  1. When graphing an inequality, you want to solve it for ‘y’ and then graph using the slope and y-intercept.
  2. Do the shading based on a ‘test point’. We often use the test point (0, 0).
  3. When you have ‘AND’ between 2 inequalities, this means that you want the shading on the graph to represent all of the points in COMMON.
  4. When you have ‘OR’ between 2 inequalities, this means that you want the shading to represent all of the points.

EXAMPLES:

Instructions: Graph the solution sets of the following linear inequalities on the graphs provided.

  1. x – 3y < 6

  1. y
  1. x + 3 > 0
  1. y > -4x – 3 OR y 3x – 5
  1. y > -4x – 3 AND y 3x – 5
  1. x > 1 OR y 4
  1. x > 1 AND y ≥ 4

Finally, let’s review how to solve and graph absolute value inequalities.

  1. Most of the time, you will have TWO answers to a math problem that involves absolute value bars. This is because the expression INSIDE of the bars can be positive OR negative and the answer will be the same. An example of this is . Because the ‘x + 3’ is inside the bars, then it can be equal to either 7 or -7 in order to equal the 7 on the right side.
  2. The absolute value bars MUST be on the LEFT side of the =, <, or > by themselves before you can take the expression out of the absolute value bars to begin solving it.

EXAMPLES:

Instructions: Solve the following absolute value equations.

  1. 2.

3. 4. -3

Instructions: Solve the following absolute value inequalities. Graph on a number line. Write in interval notation.

  1. 112. 4 +
  1. 34. -3

HOMEWORK ASSIGNMENT:

Complete in the following order:

  1. p. 118 (6, 8, 12, 14, 16, 24, 26, 28, 32)
  2. p. 230 (4, 7, 8, 16, 18, 21, 23, 24)
  3. p. 107 (26, 27, 29)
  4. p. 118 (34, 37, 38)