Chapter 4: Linear Inequalities Section 1: Using Graphs & Tables
Topics in Algebra 1 © 2001 Texas Instruments Teacher Notes 4-6
Linear Inequalities: Using Graphs & Tables Teacher Notes
Objectives
•
To illustrate how to estimate the solution set, in the real numbers, of a linear inequality using
graphical methods on a number line and on a Cartesian (x-y) graph.
•
To illustrate how to estimate the solution set, in the real numbers, of a linear inequality using
tables.
Math Highlights
In the number line method, students use a guess-and-test approach to search for the solution on
a number line. A point is chosen to test, the substitution is shown and students see whether the
statement is true or false. A point is plotted for a true statement. Students see an estimate of the
solution set built on the number line.
In the table of values method, students see a table of values for each side of the inequality. They see
where the inequality is satisfied, and then they see how to refine the estimate of the solution set.
In the x-y graphical method, students plot both sides of the inequality and use the graph to determine
the solution set for the inequality by testing points. In the 2-D graph, students are able to see which
graph is higher or lower than the other graph. This helps them estimate the solution set.
Note
: The inequalities of the form ax + b < c (for <,
≤
, >,
≥
) with a < 0 are not discussed in this
section.
Common Student Errors
•
Using graphs and tables can mislead students. They may think that they can always find the
exact solution using graphs and table. Although they will often find exact solutions using
these methods, using algebra will always give exact solutions for inequalities. For the
calculator example, x + 2
1, only integer values are tested. If the example had been
x + 2 < 1, the students would need to test points closer and closer to x =
M
1 to see that the
solution set contains values strictly less than
M
1. The endpoint (
M
1) is not included in the
solution. Encourage students to pick many points. Remind them that they would have to test
all points with these methods to get the exact solution, in the real numbers and that is
physically impossible to test all points!
•
At times, introducing the algebraic solution of inequalities gives students just the mechanics
of doing a problem. Algebraic methods alone usually do not invite students to reason out the
solution using number sense. The graphs and tables method gives students the opportunity to
see the values of each side of the inequality as a graph or table so they can compare the size
of the numbers, thus helping them create the solution set.
•
Visual learners can benefit by seeing the graphs and numbers and using them as the tool to
find the solution set.
•
Some students may still have difficulty remembering the meaning of the symbols,
<
,
≤
,
>
, and
≥
.
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