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Showing posts with label inequalities. Show all posts
Showing posts with label inequalities. Show all posts

Introduction to Inequalities and Interval Notation

All of the steps that we have learned for solving linear equations are the same for solving linear inequalities except one.  We may add or subtract any real number to both sides of an inequality and we may multiply or divide both sides by any positive real number.  The only new rule comes from multiplying or dividing by a negative number.
So whenever we divide or multiply by a negative number we must reverse the inequality. It is easy to forget to do this so take special care to watch out for negative coefficients.
Notice that we obtain infinitely many solutions for these linear inequalities.  Because of this we have to present our solution set in some way other than a big list.  The two most common ways to express solutions to an inequality are by graphing them on a number line and interval notation.

Note: We use the following symbol to denote infinity:

Tip: Always use round parentheses and open dots for inequalities without the equal and always use square brackets and closed dots for inequalities with the equal.

Systems of Linear Inequalities (Two Variables)

The systems of linear inequalities that we will be solving consist of two linear inequalities and two variables.  To solve these we will graph the solution sets of both linear inequalities and then determine where the sets intersect.  Any point in the overlap of the graphs will be a solution to the system.

Instructional Video: Systems of Linear Inequalities

Graph the solution set:
The above solution suggests that (−5, 3) is a solution because it is shaded.  Check it and others to see if it solves both inequalities.

These graphs can sometimes get messy so do your best to think about the solution before actually shading.  Use pencil and a good eraser when working on these problems.

Given the graph, determine the system.

Graph the systems of inequalities.
Graph the system:

Linear Inequalities (Two Variables)

When graphing an equation like y = 3x − 6 we know that it will be a line.  The graph of a linear inequality such as y >= 3x − 6, on the other hand, gives us a region of ordered pair solutions.

Not only do the points on the line satisfy this linear inequality - so does any point in the region that we have shaded.  This line is the boundary that separates the plane into two halves - one containing all the solutions and one that does not. Therefore, from the above graph, both (0, 0) and (−2, 4) should solve the inequality.

Use a test point not on the boundary to determine which side of the line to shade when graphing solutions to a linear inequality.  Usually the origin is the easiest point to test as long as it is not a point on the boundary.

Graph the solution set.

If the test point yields a true statement shade the region that contains it.  If the test point yields a false statement shade the opposite side.

When graphing strict inequalities, inequalities without the equal, the points on the line will not satisfy the inequality; hence, we will use a dotted line to indicate this.  Otherwise, the steps are the same.

Graph the solution set.
Given the graph determine the missing inequality.