1. 1.5 Linear Equations and Inequalities
statements that two expressions are equal
to solve an equation means to find all numbers that will satisfy the equation
the solution (or root) of an equation is said to satisfy the equation
solution set is the list of all solutions

2. 1.5 Linear Equation in One Variable
A linear equation in one variable is an equation that can be written in the form

3. 1.5 Linear Equations and Inequalities
Solving Linear Equations
analytic: paper & pencil
graphical: often supports analytic approach with graphs and tables

4. 1.5 Addition and Multiplication Properties
Addition and Multiplication Properties of Equality
For real numbers a, b, and c,

5. 1.5 Solving a Linear Equation
Example
Solve
Check

6. 1.5 Solving a Linear Equation with Fractions
Solve

7. 1.5 Graphical Solutions to f (x) = g(x)
Three possible solutions

8. 1.5 Intersection-of-Graphs Method
First Graphical Approach to Solving Linear Equations
where f and g are linear functions
set and graph
find points of intersection, if any, using intersect in the CALC menu
e.g.

9. 1.5 Intersection-of-Graphs Method
Intersection-of-Graphs Method of Graphical Solution
To solve the equation graphically, solve
The x-coordinate of any point of intersection of the two
graphs is a solution of the equation.

10. 1.5 Application
The percent share of music sales (in dollars) that compact discs (CDs) held from 1987 to 1998 can be modeled by
During the same time period, the percent share of music sales that cassette tapes held can be modeled by
In these formulas, x = 0 corresponds to 1987, x = 1 to 1988, and so on. Use the intersection-of-graphs method to estimate the year when sales of CDs equaled sales of cassettes.
Solution:
100 12

11. 1.5 The x-Intercept Method
Second Graphical Approach to Solving a Linear Equation
set and any x-intercept (or zero) is a solution of the equation

12. 1.5 The x-Intercept Method
x-intercept Method of Graphical Solution
To solve the equation graphically, solve
The x-intercept of the graph of F (or zero of the function F)
is a solution of the equation.

13. 1.5 The x-Intercept Method
Root, solution, and zero refer to the same basic concept:
real solutions of correspond to the x-intercepts of the graph

14. 1.5 Example Using the x-Intercept Method
Solve the equation
Graph hits x-axis at x = –2. Use Zero in CALC menu.

15. 1.5 Identities and Contradictions
A contradiction is an equation that has no solution.
e.g.
The solution set is the empty or null set, denoted
two parallel lines

16. 1.5 Identities and Contradictions
An identity is an equation that is true for all values in the domain.
e.g.
Solution set
lines coincide

17. 1.5 Identities and Contradictions
Note:
Contradictions and identities are not linear, since linear equations must be of the form
linear equations - one solution
contradictions - always false
identities - always true

18. 1.5 Solving Linear Inequalities
Addition and Multiplication Properties of Inequality

19. 1.5 Solving Linear Inequalities
Example

20. 1.5 Solve a Linear Inequality with Fractions
Reverse the inequality symbol when multiplying by a negative number.

21. 1.5 Graphical Approach to Solving Linear Inequalities
Intersection-of-Graphs Method of Solution of a Linear Inequality
Suppose that f and g are linear functions. The solution set of is
the set of all real numbers x such that the graph of f is above the graph of g.
The solution set of is the set of all real numbers x such that the
graph of f is below the graph of g.

22. 1.5 Intersection of Graphs Method
Example: 10
-10 10 10
-15

23. 1.5 Intersection of Graphs Method
Agreement of Inclusion of Exclusion of Endpoints for Approximations
When an approximation is used for an endpoint in specifying an
interval, we continue to use parentheses in specifying inequalities
involving < or > and square brackets in specifying inequalities
involving < or >.

24. 1.5 x-Intercept Method
x-intercept Method of Solution of a Linear Inequality
The solution set of is the set of all real numbers x such that
the graph of F is above the x-axis. The solution set of is the set
of all real numbers x such that the graph of F is below the x-axis.

25. 1.5 x-Intercept Method
Example:

26. 1.5 Three-Part Inequalities
Application
Consider error tolerances in manufacturing a can with radius of 1.4 inches.
r can vary by
Circumference varies between and r

27. 1.5 Solving a Three-Part Inequality
Example
Graphical Solution 25 25
-20 6
-20 6
-20
-20