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2.4 – Linear Equations and Inequalities

Objectives

  • Write the equation of a line in three forms.
  • Identify key components of a line from a given equation.
  • Express the solution to a linear inequality graphically.

 

Key Terms

  • Coefficients – Numbers that multiply variables.
    • The number in front of a variable
    • Can be negative or positive
    • Ex. 4x: coef is 4
    • Ex. -x: coef is -1
    • Ex. -3/5: coef is -3/5
  • Coordinates – locations on a map
    • In math, (x,y) are the coordinates that represent a point located on the Cartesian plane
    • (x,y) are in alphabetical order, and you start plotting points with the x-axis first, then the y-axis
  • Linear Inequality – An inequality in which the variable is of degree one.
    • The graph of this line is shaded above or below the line
    • The graph may or may not include the line itself (depends if the inequality has an equal bar under it or not)
  • Point-Slope Form – The equation of a straight line in the form:
    • y-{y}_{1}=m\left(x-{x}_{1}\right)
      • The x and y in the equation remain part of the equation.  Do not substitute a value for these.
      • The ({x}_{1},{y}_{1}) represent ANY point on the line.  You choose!
  • Slope-Intercept Form – A form of a linear equation that includes the slope of the line and the value of they-intercept.
    • Form:  y = mx + b
    • m:  slope (the coefficient of x)
    • b:  y-intercept (written as (0,b))
  • Standard Form of a Linear Equation – Form:  Ax + By + C = 0.
  • Undefined – A value that cannot be computed.
    • The slope of the vertical line below is undefined (ex. x=3) because it is ALL rise and NO run.

Alg 2A - Vertical Lines

  • Zero Slope – A horizontal line has no slope, also known as a zero slope (ex. y=2) because it is NO rise and ALL run.

Alg 2A - Horizontal Lines


Notes

Graphing Inequalities
  • To graph a linear inequality
    • Solve for y and shade the solution area
    • A linear equality has a border line (since it may or may not be included in the solution).
    • Ask yourself, “are the y-values above (greater than) or below (less than) the line?”  (hint: look at the inequality sign)
      • If the sign is y \geq or y \textgreater, then you shade above the line
        • All points above the line are included in the solution
      • If the sign is y \leq or y \textless, then you shade below the line
        • All points below the line are included in the solution
      • If the sign has a bar (equal to), then you use a solid line
        • This means the line is included in the solution
      • If the sign does not have a bar (not equal to), then you use a dashed line
        • This means the line is NOT included in the solution

  • Ex 1. y\leq\frac{2}{3}x+1
    • Step 1: Graph the line, but ask yourself, “Does the inequality have a bar (equal to)?”
      • If yes, graph the line as a solid line
      • If no, graph the line as a dashed line
    • Step 2: Shade the half-plane (above or below the line), but ask yourself, “Are my y-values greater or less than the line?”

Alg 2A 2.04 - Graphing Inequalities

Answer: There is a bar, so the border line is solid.  The y-coordinates are BELOW (y\leq) the line, so shade below the border line.

Review

Linear Equations in the Real World
  • Ex 1. Dianne pays $28 to enter a state fair, plus $2 for each ride. Write the equation that represents her total cost?
    • Answer:  y = 2x + 28
    • Reason: $28 is a constant price that everyone MUST pay to enter the fair.  $2 is the cost of EACH ride.  Each means multiply.  We don’t know the number of rides she will go on, so we use x to represent the number of rides.
  • Ex 2. A consultant needs to make at least $800 this week. She earns $80 for each new written piece and $40 for each review. Write an inequality that represents the possible combinations of reviews and new written pieces that she must complete?
    • Assign variables: x for written pieces and y for reviews
    • Decide on the inequality sign: “at least” means the work she does is valued “greater than or equal to” $800, so \geq800.
    • Combinations means the addition of both pieces of work (written pieces and reviews), so use a plus (+) sign for the operator
      • Answer: 80x+40y\geq800
  • Ex 3. What will the graph of y+2=\frac{1}{5}(x+1) look like?
    • Since the point-slope equation is y-y_1=m(x-x_1), and our equation has plus signs, the points are both negative (-1, -2).
    • The slope is \frac{1}{5}, so the rise is 1 and the run is 5.
    • It’s easier to create a line from slope-intercept form, so distribute and use inverse operations to convert the form.
      • y+2=\frac{1}{5}x+\frac{1}{5}
      • y=\frac{1}{5}x+\frac{1}{5}-2, convert -2 to a fraction with 5 in the denominator: -2=\frac{-10}{5}
      • y=\frac{1}{5}x+\frac{1}{5}-\frac{10}{5}
      • y=\frac{1}{5}x+\frac{-9}{5}, -9 divided by 5 = -1.8 which is easy to graph.
      • The graph would look like this:

Alg2A 02.04 Q01-1

  • Ex 4. What will the graph of y \textgreater \frac{1}{3}x-2 look like?
    • The slope is \frac{1}{3}, so the rise is 1 and the run is 3.
    • The y-intercept is -2. Start there, plot a point at (0,-2).
    • Follow the slope up 1 and right 3, plot another point.
    • Since there is NO bar under the inequality sign, connect the points with a DASHED line.
    • Since y \textgreater, shade ABOVE the border line.
    • The graph would look like this:

Alg2A 02.04 Q01-7

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