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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 xaxis first, then the yaxis
 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)
 PointSlope Form – The equation of a straight line in the form:

 The x and y in the equation remain part of the equation. Do not substitute a value for these.
 The represent ANY point on the line. You choose!
 SlopeIntercept Form – A form of a linear equation that includes the slope of the line and the value of theyintercept.
 Form: y = mx + b
 m: slope (the coefficient of x)
 b: yintercept (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.
 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.
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 yvalues above (greater than) or below (less than) the line?” (hint: look at the inequality sign)
 If the sign is or , then you shade above the line
 All points above the line are included in the solution
 If the sign is or , 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.
 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 halfplane (above or below the line), but ask yourself, “Are my yvalues greater or less than the line?”
Answer: There is a bar, so the border line is solid. The ycoordinates are BELOW () 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 .
 Combinations means the addition of both pieces of work (written pieces and reviews), so use a plus (+) sign for the operator
 Answer:

 Ex 3. What will the graph of look like?
 Since the pointslope equation is , and our equation has plus signs, the points are both negative (1, 2).
 The slope is , so the rise is 1 and the run is 5.
 It’s easier to create a line from slopeintercept form, so distribute and use inverse operations to convert the form.
 , convert 2 to a fraction with 5 in the denominator:
 , 9 divided by 5 = 1.8 which is easy to graph.
 The graph would look like this:

 Ex 4. What will the graph of look like?
 The slope is , so the rise is 1 and the run is 3.
 The yintercept 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 , shade ABOVE the border line.
 The graph would look like this:

Important!
Practice (Apex Study 2.4)
 Practice: Pgs 3, 8, 11, 14, 15, 18, 21
 Watch the animation information on Pg 20
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