Like with systems of linear equations, we can solve linear inequalities by graphing.


 Gertrude Green
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1 Like with systems of linear equations, we can solve linear inequalities by graphing. Section 1: Reminder on Graphing Word Bank smaller equation slope slopeintercept yintercept bigger The of a line looks like this: y = mx + b. The above form of a line s equation is called m represents the slope b represents the yintercept form, where: is the steepness of a line. If a line is very steep, the slope is. However, if a line is not very steep (more flat), the slope is. is where a line crosses the yaxis.  Think of it as where the line intercepts (or meets with) the yaxis. Example: The line y = 1 x 1 has a slope of and a yintercept of. 2 To graph this line, we start by placing a point on the yintercept. This point should be at (0,1). Next, we will use the slope of the line to find additional points on the line. If a line has a slope of 1, then we know that from the location of our first plotted 2 point, we must: rise: space(s) and run: space(s) Note: when we say run on a graph, we mean: move to the (right / left).
2 Example 2: For a line with a slope of 3 we would rise: space(s) and run: space(s). 4 That means, move space(s) (up / down). And, move space(s) (right / left). Example 3: For a line with a slope of 1 2 That means, move space(s) (up / down). we would rise: space(s) and run: space(s). And, move space(s) (right / left). Note: we rise a (positive / negative) amount. Example 4: For a line with a slope of 3 we would rise: space(s) and run: space(s). 4 That means, move space(s) (up / down). And, move space(s) (right / left). Note: we run a (positive / negative) amount. Careful on this one. Since we are running a negative amount, we will actually be moving to the left, not the right (Note: yes, I realize I just answered the previous blank for you thanks for noticing!) Example 5: For a line with a slope of  4 the negative sign can go in either the numerator (top) or the 5 denominator (bottom). If we put it in both parts, however, we would end up with 4 and when you divide a negative by a negative like this, you will end up with a positive. That is to say, 4 5 is the exact same thing as 4 5. And 4 5 is equal to 4 5. So, we will rise: space(s) and run: space(s). 5 Example 6: For a line with a slope of 2 we would rise: space(s) and run: space(s). Note: 2 is equal to 2 1
3 Practice: On the coordinate plane to the left, graph the line y = 4x +2. Your yintercept will be at. So, your first point should be at: (, ) To find your next point, you will rise: space(s) and run: space(s). Repeat this procedure to find additional points. In the practice problem above, complete the line on your graph so that it crosses the entire coordinate plane. For the next section of this activity, it will be important that you do this when graphing all lines. For instance, the graph of y = 1 3 x 2 should look like this: Now, consider all the points that are less than the line y = 1 3 x 2 Do you think you will find these points above or below the line? (above / below) With a colored pencil or highlighter, color all of the points on the above graph that are less than the line y = 1 x 2. To represent all 3 of these points, along with the line, we will write y 1 3 x 2. Practice: On the axis to the right, graph y 2x 4 Did you color your graph above or below the line?
4 What is the difference between y 2x 4 and y < 2x 4? Section 2: Graphing Inequalities If we wanted to make a graph of all of the points that are less than y = 2x 4, without including the line, we would write y < 2x 4. In this case, we would draw a dotted line instead of a regular line, and then color beneath the dotted line. It should look like this: Practice: Now, you try. On the coordinate plane below, graph y < 1 x What is the difference between y < 1 x + 2 and y > 1 x + 2? 3 3 Where do you think you should color when y is greater than 1 x + 2? Above or Below the line: 3 Now, on the coordinate plane below, graph y > 1 3 x + 2 What is the difference between y > 1 3 x + 2 and y 1 3 x + 2? Which one has a dotted line? Which one has a complete line?
5 Section 3: Graphing Systems of Inequalities Word Box Intersection Ordered Pair Divide Standard SlopeIntercept Common Two or More Equations A system of equations is a set of two or more. When we were solving systems of linear equations, we graphed both lines and found the. That is to say, the place where the two lines cross (a.k.a. the point that the two lines have in ). A system of inequalities is a set of inequalities. Just like with systems of equations, we can use graphing to solve a system of inequalities. Practice: On the coordinate plane to the right, graph both of these inequalities: y 3x 1 y 2x + 2 (Use different colors to fill in the colored areas of your graph). Next, in dark pencil or pen, outline the areas that the two graphs have in common. That is the solution to your system of inequalities the part your two graphs have in common. Now, on the following grid, graph your two inequalities, and only color the parts that they have in common.
6 Practice For the following systems of inequalities, graph to find the solution. y > x + 2 y < 2x + 3 x + y 2 y < 1 4 x  1 y x 3 y < 4x + 5 y 1 2 x + 4 y > 2x 3 Note: If you are given an inequality in the form 3x 4y > 1 which is in form, you must first put the inequality into form by solving for y. Example: 3x 4y > 1 To solve for y, I want y by itself, so move everything else to the other side of the inequality. 3x 4y > 13x 3x (Subtract 3x from both sides of the inequality)  4y > 3x + 14y > 3x + 1 (Divide both sides by 4) 44 y < 3 x + 1 (Note: the > becomes < when you by a negative number. 4 4 y < 3 4 x 1 4
The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6
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