1. Algebra 1
Section 6.7

2. Linear Inequalities
Linear inequalities are similar to linear equations but contain an inequality sign instead of an equals sign.
Inequalities use ≠, <, >, ≤, or ≥ and have an infinite number of solutions.

3. Linear Inequalities
The graph of a linear equation, such as x = 3 or 2x + y = 3, separates the coordinate plane into three distinct sets of points.
x = 3; x < 3; x > 3
2x + y = 3; 2x + y < 3; 2x + y > 3

4. Example 1 Graph 2x + y > 3. y > -2x + 3 Slope: -2
y-intercept: (0, 3)

5. Example 1
Since y > -2x + 3 is an inequality, not an equation, the solution set does not include the points on the line. Graph the equation y = -2x + 3 with a dashed line to represent this boundary.

6. Example 1
We need to determine which side of the line contains the solutions to the inequality.
The origin, (0, 0), is an easy choice.

7. Example 1
Substitute the x and y values of the point into the inequality.
2(0) + 0 > 3, so (0, 0) is not a solution.
The solutions must be on the other side of the line.

8. Example 1
Shade the side of the line that contains the solution set.

9. Graphing an Inequality
Solve for y and identify the slope and y-intercept of the corresponding equality.
Graph the corresponding linear equation.

10. Graphing an Inequality
If the inequality is < or >, use a dashed line to indicate that points on the line are not solutions.
If the inequality is ≤ or ≥, use a solid line to indicate that the points are solutions.

11. Graphing an Inequality
Choose a test point that is not on the line and substitute it into the inequality.

12. Graphing an Inequality
If the resulting inequality is true, shade the side containing the test point.
If the inequality is false, shade the other side.

13. Example 2 Solve for y and identify the slope and y-intercept:
y ≤ 3x + 5
Slope: 3
y-intercept: (0, 5)

14. Example 2
Since the inequality sign is ≤, graph y = 3x + 5 with a solid line.
(0, 0) is a solution to the inequality, so shade the side of the line containing that point.

15. Example 2

16. Linear Inequalities
Linear inequalities are often used to model real-life situations in which there are acceptable ranges of values.
The domain and range are often limited to nonnegative numbers.

17. Example 3 Let x = grams of carbohydrates y = grams of fats
total calories ≤ 720
calories from carbohydrates = 4x
calories from fat = 9y
4x + 9y ≤ 720

18. Example 3
Graph 4x + 9y ≤ 720 with a solid line. It contains the x-intercept (180, 0) and y-intercept (0, 80).
Since she cannot consume negative amounts, only the first quadrant and axes need to be considered.

19. 4(100) + 9(30) ≤ 720, so this fits her diet.
Example 3
100 g carbohydrates, 30 g fats.
4(100) + 9(30) ≤ 720, so this fits her diet.

20. Homework: pp