1. 2.4 – Linear inequalities and problem solving

2. Objectives I will use interval notation.
I will solve linear inequalities using the addition property of inequality.
I will solve linear inequalities using the multiplication property of inequality.
I will solve problems that can be modeled by linear inequalities.

3. Why are linear inequalities important?
Suppose a salesperson earns a base salary of $600 per month plus a commission of 20% of sales. Find the minimum amount of sales needed to receive a total of at least $1500 per month. Here, the phrase “at least” implies that an income of $1500 or more is acceptable. In symbols, we can write…
income ≥ 1500

4. Linear Inequality in one variable
A linear inequality in one variable is an inequality that can be written in the form
ax + b < c
Where a, b, and c are
real numbers and a ≠ 0

5. Solution
A solution of an inequality is a value of the variable that makes the inequality a true statement.

6. Solution Set
The solution set of an inequality is the set of all solutions.
x > 2
Contains all numbers greater than 2
The graph of {x/ x>2} looks like the following. 2

7. Interval Notation
With interval notation use a parenthesis instead of an open circle. The graph of {x/ x>2} will now look like this…
This graph can be represented in interval notation as (2, ∞) The graph parenthesis indicates that 2 is NOT included in the interval. 2

8. Interval Notation
Use a bracket if a number IS included in the interval. To graph, {x / x ≥ 2} use a bracket instead of a parenthesis.
This inequality written in set notation is [2, ∞) 2

9. Check your understanding
Set Notation: {x / x < a}
Graph:
Interval Notation: (-∞, a)
See page 84 a

10. Example 1
Graph each set on a number line and then write in interval notation.
a. {x/ x ≥ 2} b. {x / x < -1} c. {x/ 0.5 <x ≤3

11. Addition Property of Inequality
If a, b, and c are real numbers, then
a < b and a + c < b + c

12. Example 2 x – 2 < 5 x – 2 + 2 < 5 + 2 x < 7
Solve x -2 < 5. Graph the solution set.
x – 2 < 5
x – < 5 + 2
x < 7
The solution set is {x / x < 7}
Interval notation is (- ∞, 7) 7 8 6

13. You try it! Write the solution in set notation, interval notation, and graph
Solve: x +3 < 1

14. Solution Set: {x / x ≥ -10} Interval Notation: [-10, ∞)
Example 3
Solve 3x + 4 ≥ 2 x – 6
3 x + 4 ≥ 2 x – 6
3 x + 4 – 4 ≥ 2x – 6 – 4
3x ≥ 2x – 10
3x – 2x ≥ 2x – 10 – 2x
x ≥ -10
Solution Set: {x / x ≥ -10} Interval Notation: [-10, ∞) [
-10 -9
-11

15. You try it!
Solve: 5 x – 1 ≥ 4 x + 4

16. Multiplication property of inequality
If a, b, and c are real numbers and c is positive then
a < b and ac < bc
are equivalent inequalities.
If a,b,c and c are real numbers and c is negative, then
a < b and ac > bc

17. Hint
Whenever both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality symbol must be reversed to form an equivalent inequality.

18. Example 4: Solve and graph the solution set.
Multiply both sides
by 4
Inequality symbol is the same since
we are multiplying by a positive
number.

19. Example 4: Solve and graph the solution set.
x > -3
Set Notation {x/ x > -3} Interval Notation (-3, ∞)
Inequality symbol
is reversed since
we divided by a
negative number.

20. -2x > 7 2x – 3 > 10 -x + 4 + 3x < 7 - x + 4 < 5
Concept Check - In which of the following inequalities must the inequality symbol be reversed during the solution?
-2x > 7
2x – 3 > 10
-x x < 7
- x + 4 < 5

21. Steps for solving a linear inequality in one variable
Step 1: Clear the equation of fractions
Step 2: Use distributive property to remove grouping symbols.
Step 3: Combine like terms on each side of the inequality.
Step 4: Use the addition property of inequality
Step 5: Use the multiplication property of inequality.
ULTIMATE GOAL- ISOLATE THE VARIABLE!

22. Example 5 Solve: - (x – 3) + 2 ≤ 3 (2x – 5) + x
-x x ≤ 7 x – 15 + x
20 ≤ 8 x
Write the problem
Distributive property
Combine like terms
Add x to both sides
Add 15 to both sides
Divide both sides by 8 and
simplify

23. You try it! 
-(2x – 6) ≤ 4 (2x – 4) + 2

24. Example 6: Solve: Write inequality Clear the fractions Distribute
Subtract 5 x from both sides
Add 12 to both sides
Divide both sides by negative
3, this will reverse inequality
symbol!

25. You try it! 

26. Example 7: Solve 2 (x+3) > 2x + 1
2x + 6 – 2x > 2x + 1 – 2x
6 > 1
6 >1 is a true statement for all values of x, so this inequality and the original inequality are true for all numbers.
{x / x is a real number} OR (-∞ , ∞)

27. You try it! 
5(x – 3) < 5x + 2

28. Example 8 – Calculating income with commission
A salesperson earns $600 per month plus a commission of 20% of sales. Find the minimum amount of sales needed to receive a total income of at least $1500 per month.
TRANSLATE: Let x = amount of sales
x ≥ 1500

29. Interpret: The income for sales of $4500 is
Solve
x ≥ 1500
x ≥ – 600
0.20x ≥ 900
x ≥ 4500
Interpret: The income for sales of $4500 is
(4500) = 1500
Thus, if sales are greater than or equal to $4500, income is greater than or equal to $1500.

30. Try it! 
A salesperson earns $1000 a month plus a commission of 15% of sales. Find the minimum amount of sales needed to receive a total income of at least $4000 per month.

31. Example 9 – Finding annual consumption
In the U.S., the annual consumption of cigarettes is declining. The consumption c in billions of cigarettes per year since the year 1985 can be approximated by the formula
c = t
Where t is the number of years after Use this formula to predict the years that the consumption of cigarettes will be less than 200 billion per year.

32. Understand
Read and re-read the problem. To become familiar with the given formula, let’s find the cigarette consumption after 25 years, which would be the year , or To do so, we substitute 25 for t in the given formula.
c = (25)
c =
Thus, in 2010, we predict cigarette consumption to be about billion.

33. Translate
We are looking for the years that consumption of cigarettes c is less than Since we are finding years t, we substitute the expression in the formula for c, or
t < 200

34. Solve: -14.25 t + 598.69 < 200 -14.25 t + 598.69 < 200

35. Interpret:
We substitute a number greater than and see that c is less than 200.
The annual consumption of cigarettes will be less than 200 billion for the years more than years after 1985, or approximately = 2013.

36. You Try It! 
Use the formula c = t to predict when the consumption of cigarettes will be less than 100 billion per year.