Graphing+Linear+Inequalities

Solving Linear Inequalities Page Written By: Thomas Micah Gestrich Video Example on how to graph linear inequalities: media type="youtube" key="0VVQYOMRlkQ" height="344" width="425" Originally from: http://www.yourteacher.com/

=Graphing Linear Inequalities=

In this lesson, we will learn how to solve inequalities using graphs. In the following diagram: Aall the points above the line //y// = 1 are represented by the inequality //y// > 1. All the points below the line are represented by the inequality //y// < 1. The representation is clearer if you look at what the //y//-coordinates of these points have in common. In the diagram below, the region above the line is represented by //y// > 2//x// –1 and the region below the line is represented by //y// < 2//x// – 1.



//**Example:**// By shading the unwanted region, show the region represented by the inequality 2//x// – 3//y// ≥ 6 //**Solution:**// First, we need to draw the line 2//x// – 3//y// = 6.

We will revise the method for drawing a straight line. Rewrite the equation in the form //y// = //mx + c.// From the equation //m// will be the gradient and //c// will be the //y//-intercept. 2//x// – 3//y// = 6 ⇒ //y// = //x// – 2 The gradient is then and the //y//-intercept is – 2. If the inequality is ≤ or ≥ then we draw a solid line. If the inequality is then we draw a dotted line. After drawing the line, we need to shade the unwanted region. Rewrite the inequality 2//x// – 3//y// ≤ 6 as //y// ≥ //x// – 2. Since the inequality is ≥, the wanted region is above the line and so the unwanted region is below the line. We shade below the line.

//**Example:**// By shading the unwanted region, show the region represented by the inequality //x + y// < 1 //**Solution:**// Rewrite the equation //x + y// = 1in the form //y// = //mx + c.// //x + y// =1 ⇒ //y// = –//x// + 1 The gradient is then –1 and the //y//-intercept is 1. We need to draw a dotted line because the inequality is <. After drawing the dotted line, we need to shade the unwanted region. Rewrite the inequality //x + y// < 1 as //y// < –//x// + 1. Since the inequality is <, the wanted region is below the line and so the unwanted region is above the line. We shade above the line.



__Graphing Linear Inequalities__. Onlinemathtutoring.com. 14 Nov. 2008 .

Once you've learned how to [|graph linear inequalities] , you can move on to solving systems of linear inequalities. A "system" of linear inequalities is a set of linear inequalities that you deal with all at once. Usually you start off with two or three linear inequalities. The technique for solving these systems is fairly simple. Here's an example. **2//x// – 3//y//** __<__ **12** >  **//x// + 5//y//** __<__ **20**     * **//x// > 0** Just as with solving single linear inequalities, it is usually best to solve as many of the inequalities as possible for " //  y  //  " on one side. Solving the first two inequalities, I get the rearranged system: //y// __>__ ( 2/3 )//x// – 4 >  //y// __<__ ( – 1/5 )//x// + 4   *   //x// > 0     Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved   "Solving" systems of linear inequalities means "graphing each individual inequality, and then finding the overlaps of the various solutions". So I graph each inequality, and then find the overlapping portions of the solution regions. ||  ||   || > ||  This inequality is a "greater than" inequality, so I want to shade above the line. However. since there will be more than one inequality on this graph, I don't know (yet) how much of that upper side I will actually need. Until I know, I can keep track of the fact that I want the top region by drawing a little "fringe" along the top side of the line, like this: > || The last inequality is a common "real life" constraint: only allowing   //x//   to be positive. The line  "//x// = 0"   is just the   //y//   -axis, and I want the right-hand side. I need to remember to dash the line in, because this isn't an "or equal to" inequality, so the boundary (the line) isn't included in the solution: > || The "solution" of the system is the region where all the inequalities are happy; that is, the solution is where all the inequalities work, the region where all three individual solution regions overlap. In this case, the solution is the shaded part in the middle: > **2//x// – //y// > –3** >  **4//x// + //y// < 5**     As usual, I first want to solve these inequalities for "  //  y  //  ". I get the rearranged system: //y// < 2//x// + 3 > //y// < –4//x// + 5 ||  ||   ||  The kind of solution displayed in the above example is called "unbounded", because it continues forever in at least one direction (in this case, forever downward).
 * //**Systems of Linear Inequalities**//
 * ** Solve the following system:  **
 * The solution region for the [|previous example]  is called a "closed" or "bounded" solution, because there are lines on all sides. That is, the solution region is a bounded geometric figure (a triangle, in that case). You can also obtain solutions that are "open" or "unbounded"; that is, you will have some exercises which have solutions that go off forever in some direction. Here's an example:
 * **Solve the following system:**

Of course, there's always the possibility of getting no solution at all. For instance: > ** Solve the following system:  **  Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved **//x// – //y//** __<__ **–2** > **//x// – //y//** __>__ **2** First I solve for //  y  // , and get the equivalent system: //y// __>__ //x// + 2 > //y// __<__ //x// – 2 But there is no place where the individual solutions overlap. (Note that the lines  //y// = //x// + 2   and   //y// = //x// – 2   never intersect, being parallel lines with different  //  y  //  -intercepts.) Since there is no intersection, there is   **no solution.** **__Stapel, Elizabeth. "Systems of Linear Inequalities." Purplemath. Available from http://www.purplemath.com/modules/syslneq2.htm. Accessed 05 November 2008__** **__Definitions:__**

__Open half-planes__- The coordinated plane is seperated into two regions

__Closed half-plane__- "closed" means the boundary line is included


 * __Practice Problems:__**

http://hotmath.com/help/gt/genericalg1/section_4_4.html


 * __Interactive Programs to help out:__**

http://www.mhhe.com/math/devmath/aleks/wt-ia/student/olc/graphics/author_ed/chp3sec23.htm


 * __Practice Test Problems:__**

http://www.templejc.edu/dept/Math/RSimpson/Math0303/303PracticeTest1.pdf


 * __Graphing Linear Inequalities Powerpoint__**

http://www.lz95.org/lzhs/math/faculty/Bivin/Power%20point%20math/graphing_linear_inequalities.ppt. http://www.algebralab.org/studyaids/studyaid.aspx?file=Algebra2_2-6.xml http://www.mathwarehouse.com/algebra/linear_equation/linear-inequality.php http://www.math.tamu.edu/~pivarski/Fall2006/141week5part1.pdf
 * __Relative links related to this Topic:__**

Quiz Questions:


 * Graph the solution set of each system in a coordinated plane.**

1. //x// < 2 //y// > -1

2. //x// __<__ -//y// 2y < //x//

3. -2//y// > //x// - 8 4//y// > //x// - 12

4. 2//x// + //y// < 2 //x// - //y// > -2 //x// + //y// > -2

5. //x// > -2 //y// > 0 2//y// < //x// + 4 //y// < -3//x// + 9