Objective:

  • Learn to solve graph with Inequalities


Point on a Graph

The coordinate plane is a grid with an x-axis and a y-axis. 

 

Every point on a plane is an ordered pair, which consists of two numbers separated by a comma and enclosed by parentheses (x, y). The first number is the x value and the second number is the y value.  Example: (-4, 7) is the point on the coordinate plane where x = -4 and y = -7. 

 

The point where x-axis and y-axis meet is called the origin and has the ordered pair (0, 0).

 

The x value in the ordered pair is the value on the right (if positive) or to the left (if negative) from the origin (0,0).

 

The y value in the ordered pair is the value on the up (if positive) or to the down (if negative) from the origin (0,0).

 

 

 

Exercise 1: Draw the point on the graph.


Lines and Equations

Lines that cross the x-axis and do not cross the y-axis have equations that contain only the variable x and not the variable y. For example x =  3.

Lines that cross the y-axis and do not cross the x-axis have equations that contain only the variable y and not the variable x. For example y = 7.

 

 

Exercise 2: Draw the graph for given equation.


Slope of a Line

When the two or more points on a plane form a line, a special equation, called linear equations, can be written to represent all of the points on the line.

 

Line that cross both the x-axis and the y-axis have equations that contain both the variables x and y. For example, y = x + 1

 

 

Every line has a slope, except vehicle line (have no slope).

Slope is found by comparing the positions of any two points on the line.

For example, for a given two points (x1, y1) & (x2, y2), the slope of the line would be (y1-y2) / (x1 - x2). It is also described as (rise)/(run) or (the change in y)/the change in x).

 

Ex: If two points on a line are given as (-2,0) and (1, 02) then the slope of the line is  (y1-y2)/(x1-x2) =( 0-(2))/(-2-1) = 2/(-3) = -(2/3)

 

PARALLEL: If two lines are parallel then the slopes of both lines are parallel.

PERPENDICULAR: If two lines are perpendicular then the slopes of perpendicular lines will be negative reciprocals. For example, if the slope of the line 1 is m1 and the slope of line 2 is m2, and the line l1 and l2 are perpendicular then m1 = - (1/m2) or (m1)(m2) = -1.

 

Exercise 3: Draw the graph for given inequality equation.


Graphing Lines

The equation for a line is y=mx +b where m is the slope of the line and b is the y-intercept (y-value of the point where the line intersects y-axis).

 

Case I: The Horizontal lines have equations which simplify to the form y = b, where b is the y-intercept. The slope of these lines is zero.

Case II: The Vertical lines have no sope and the equation of the line takes the form of x=c, where c is the x-intercept.

Case III: Neither horizontal nor vertical line:

1. If the slope and y-intercept value is given then substitute these numbers in the slope-intercept form os a linear equation, y= mx+b, where m is the slope and b is the y-intercept.

2. If the slope (m) and a point (x1, y1)  is  given then solve for equation y = mx +b

   since m = (y -y1)/(x - x1) => y - y1 = m (x - x1) => y - y1 = mx - mx1 => y = mx - mx1 + y1  => y = mx + (y1 - mx1)

3. Given 2 points (x1, y1) and (x2, y2) on the line, then solve for the equation y = mx + b

       Step 1: Find the slop, m =(y2-y1) / (x2 -y1) to form the equation y = mx + b

       Step 2: Find the value of b, by substituting the (x1, y1) and (x2, y2) values in the equation y = mx + b.

4. Given the equation of another line:

      Option 1: To find the equation for the parallel line, the slop of the equation will the same as the slope of the given line..

      Option 2: To find the equation for the perpendicular line, the slope of the line will be the negative reciprocal of the given slope.

Exercise 4: Draw the graph for given inequality equation.


Graphs of Inequalities

x > n means that any number x that is greater than n is number x is in the solution set . For example, for a given equation x > 3 means that all real numbers that are greater than 3 such as  4, 5, 6 are in the solution set. However, numbers such as 1, 2 and 3 are not in the solution set.

 

Examples:

1.)    x > -2 means all real numbers greater than or equal to –2.

2.)    x< 2 means all real numbers less than 2.

3.)    x< 2  means all real numbers less than or equal to 2.

Example: Draw the graph for equation y < 7

 

 

 

Exercise

Exercise 5: Draw the graph for given inequality equation


 

Simple Inequalities

A simple inequality is solved basically the same way as a first-degree equation. The only difference is when multiplying or dividing by a negative number, switch the inequality sign.

Examples:

    -7 > -10x + 13

    => -7 -13 > -10x +13 -13

    => -20 > -10x

    => -2 > -x

    =>  2 < x

 

Exercise

Exercise 6: Simplify the inequality equation.

Exercise 7: Draw the graph for the given inequality equation..


 

Compound Inequalities & Conjunction 

Conjunction: “and” For a conjunction to be true both parts of the inequality must be true. The graph is the intersection of both solution sets.

 

Example: Solve x > -2 and x < 3

(x > -2)

 

 

(x < 3)

 

 

(-2 < x < 3)

 

 

 

Solution: “The set of all real numbers x that x is greater than –2 and less than 3”.

 

Exercise

Exercise 8: Draw the graph based on the inequality equations with conduction 


Compound Inequalities & Disjunction 

Disjunction: “or” For a disjunction to be true at least one part of the inequality must be true (both parts may be true). The graph is the union of both solution sets.

Example: Solve x > -2 or x < 3

 

(x > -2)

 

 

(x < 3)

 

 

(-2 < x < 3)

 

 

 

Solution: “The set of all real numbers such that x is greater than –2 or  less than 3”.

Exercise

Exercise 9: Draw the graph based on the inequality and disjunction.


Review

If x>n, then a  number (x) greater than  number (n) will be with in the solution set, and all the other value of x will not be in the solution set.


Lessons Learned

  • Identifying values that are within the set based on the given  inequality equation.


Quiz

None


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