1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§4.1 Solve InEqualities
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 2.5 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
MTH 55
Review §
Any QUESTIONS About
§2.5 → Point-Slope Line Equation
Any QUESTIONS About HomeWork
§2.5 → HW-7

3. Solving InEqualities An inequality is any sentence containing
Some Examples
ANY value for a variable that makes an inequality true is called a solution. The set of all solutions is called the solution set. When all solutions of an inequality are found, we say that we have solved the inequality.

4. Linear InEqualities
A linear inequality in one variable is an inequality that is equivalent to one of the forms that are similar to mx + b
where a and b represent real numbers and a ≠ 0.

5. Example  Chk InEqual Soln
Determine whether 5 is a solution to
a) 3x + 2 >7 b) 7x − 31 ≠ 4
SOLUTION
Substitute 5 for x to get 3(5) + 2 > 7, or 17 >7, a true statement. Thus, 5 is a solution to InEquality-a
Substitute to get 7(5) − 31 ≠ 4, or 4≠ 4, a false statement. Thus, 5 is not a solution to InEquality-b

6. “Dot” Graphs of InEqualities
Because solutions of inequalities like x < 4 are too numerous to list, it is helpful to make a drawing that represents all the solutions
The graph of an inequality is such a drawing. Graphs of inequalities in one variable can be drawn on a number line by shading all the points that are solutions. Open dots are used to indicate endpoints that are not solutions and Closed dots are used to indicated endpoints that are solutions

7. Example  Graph InEqualities
Graph InEqualities: a) x < 3, b) y ≥ −4; c) −3< x ≤ 5
Soln-a) The solutions of x < 3 are those numbers less than 3.
Shade all points to the left of 3
The open dot at 3 and the shading to the left indicate that 3 is NOT part of the graph, but numbers such as 1 and −2 are

8. Example  Graph InEqualities
Graph Inequalities: a) x < 3, b) y ≥ −4; c) −3< x ≤ 5
Soln-b) The solutions of y ≥ −4 are shown on the number line by shading the point for –4 and all points to the right of −4.
The closed dot at −4 indicates that −4 IS part of the graph

9. Example  Graph InEqualities
Graph InEqualities: a) x < 3, b) y ≥ −4; c) −3< x ≤ 5
Soln-c) The inequality −3 < x ≤ 5 is read “−3 is less than x, AND x is less than or equal to 5.”
Note the
OPEN dot at −3 → due to −3< x
CLOSED dot at 5 → due to x≤5

10. Interval Notation
Interval Notation for Inequalities on Number lines can used in Place of “Dot Notation:
Open Dot, ס → Left or Right, Single Parenthesis
Closed Dot, ● → Left or Right, Single Square-Bracket

11. Interval vs Dot Notation
Graph x ≥ 5
Dot Graph
Interval Graph [
Graph x < 2
Dot Graph
Interval Graph )

12. Interval Graphing of InEqualities
If the symbol is ≤ or ≥, draw a bracket on the number line at the indicated number. If the symbol is < or >, draw a parenthesis on the number line at the indicated number.
If the variable is greater than the indicated number, shade to the right of the indicated number. If the variable is less than the indicated number, shade to the left of the indicated number.

13. Set Builder Notation
In MTH55 the INTERVAL form is preferred for Graphing InEqualities
A more compact alternative to InEquality Solution Graphing is SET BUILDER notation:
SET BUILDER Notation
Have gone over this in past lectures
Read as: “x such that x is…

14. Compact Interval Notation
Graphed Interval Notation can be written in Compact, ShortHand form by transferring the Parenthesis or Bracket from the Graph to Enclose the InEquality.
Examples
x  13 → (−, 13 ]
−11< x  13 → (−11, 13]
−11< x → (−11, )

15. Example  SetBuilder & Interval
Write the solution set in set-builder notation and interval notation, then graph the solution set.
a) x ≤ −2 b) n > 3
SOLUTION a)
Set-builder notation: {x|x ≤ −2}
Interval notation: (−, −2]
Graph ]

16. Example  Set Builder
Write the solution set in set-builder notation and interval notation, then graph the solution set.
a) x ≤ −2 b) n > 3
SOLUTION b)
Set-builder notation: {n|n > 3}
Interval notation: (3, )
Graph (

17. Intervals on the Real No. Line

18. Addition Principle for InEqs
For any real numbers a, b, and c:
a < b is equivalent to a + c < b + c;
a ≤ b is equivalent to a + c ≤ b + c;
a > b is equivalent to a + c > b + c;
a ≥ b is equivalent to a + c ≥ b + c;
Example: 3 < 7 => (3+5) < (7+5)

19. Example  Addition Principle
Solve & Graph
Solve (get x by itself)
Addition Principle
Simplify to Show Solution
Graph (
Any number greater than −4 makes the statement true.

20. Multiplication Principle for InEqs
For any real numbers a and b, and for any POSITIVE number c:
a < b is equivalent to ac < bc, and
a > b is equivalent to ac > bc
For any real numbers a and b, and for any NEGATIVE number c:
a < b is equivalent to ac > bc, and
a > b is equivalent to ac < bc
Similar statements hold for ≤ and ≥
Do example: Mult: 11>7 by -3 to obtain -33 < -21

21. Mult. Principle Summarized
Multiplying both Sides of an Inequality by a NEGATIVE Number REVERSES the DIRECTION of the Inequality
Examples

22. Example  Solve & Graph Solve & Graph a) b) Soln-a) Graph
Divide Both Sides by −4
Reverse Inequality as the Eqn-Divisor is NEGATIVE
Graph
Chk with y=0, and y=3 (
The Solution Set: {y|y > −5}

23. Example  Solve & Graph Soln-b) Graph ] The Solution Set: {x|x ≤ 28}
Multiply Both Sides by 7
Simplify
Graph ]
The Solution Set: {x|x ≤ 28}

24. Example  Add & Mult Principles
Solve & Graph
SOLUTION
Add ONE to Both sides
Simplify
Subtract x from Both Sides
Divide Both Sides by 3
Simplify & Show Solution ]

25. Example  Solve 3x − 3 > x + 7
Soln
Add 3 to Both Sides
Simplify
Subtract x from Both Sides
Divide Both Sides by 2
Simplify
Graph ( - - - - - - 2 2 2 2 2 2 - - - 1 1 1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 8 8 8
The Solution Set: {x|x > 5}

26. Example  Solve 15.4 − 3.2x < −6.76
Soln
To Clear Decimals
Dist. in the 100
Simplify
Subtract 1540
Simplify; Mult. By −1/320
Simplify; note that Inequality REVERSED by Neg. Mult.
The Solution Set: {x|x > 6.925}

27. Example  Solve & Graph Solve Soln
Use Distributive Law to Clear Parentheses
Simplify
Add 3 to Both Sides
Simplify
Add 2x to Both Sides
Simplify; Divide Both Sides by 6

28. Example  Solve & Graph Solve Soln Graph The Solution Set: {x|x ≤ –2}.
From Last Slide
Put x on R.H.S.; Note Reversed Inequality
Graph ] - - - - - - 8 8 8 8 8 8 - - - 7 7 7 - - - 6 6 6 - - - 5 5 5 - - - 4 4 4 - - - 3 3 3 - - - 2 2 2 - - - 1 1 1 1 1 1 2 2 2 2 2 2
The Solution Set: {x|x ≤ –2}.

29. Equation ↔ Inequality Equation Replace = by Inequality x = 5 <≤
3x + 2 ≤ 14
5x + 7 = 3x + 23
>
5x + 7 > 3x + 23
x2 = 0 ≥
x2 ≥ 0

30. Terms of the (InEquality) Trade
An inequality is a statement that one algebraic expression is less than, or is less than or equal to, another algebraic expression
The domain of a variable in an inequality is the set of ALL real numbers for which BOTH SIDES of the inequality are DEFINED.
The solutions of the inequality are the real numbers that result in a true statement when those numbers are substituted for the variable in the inequality.

31. Terms of the (InEquality) Trade
To solve an inequality means to find all solutions of the inequality – that is, the solution set.
The solution sets are intervals, and we frequently graph the solutions sets for inequalities in one variable on a number line
The graph of the inequality x < 5 is the interval (−, 5) and is shown here ) 5
x < 5, or (–∞, 5)

32. Terms of the (InEquality) Trade
A conditional inequality such as x < 5 has in its domain at least one solution and at least one number that is not a solution
An inconsistent inequality is one in which no real number satisfies it.
An identity is an inequality that is satisfied by every real number in the domain.

33. THE NON-NEGATIVE IDENTITY
for ANY real number x
Because x2 = x•x is the product of either (1) two positive factors, (2) two negative factors, or (3) two zero factors, x2 is always either a positive number or zero. That is, x2 is never negative, or is always nonnegative

34. Solving Linear InEqualities
Simplify both sides of the inequality as needed.
Distribute to clear parentheses.
Clear fractions or decimals by multiplying through by the LCD just as was done for equations. (This step is optional.)
Combine like terms.
Use the addition principle so that all variable terms are on one side of the inequality and all constants are on the other side. Then combine like terms.
Use the multiplication principle to clear any remaining coefficient. If you multiply (or divide) both sides by a negative number, then reverse the direction of the inequality symbol.

35. Example  Solve InEquality
Solve 8x + 13 > 3x − 12
SOLUTION
8x − 3x + 13 > 3x − 3x − 12
Subtract 3x from both sides.
5x + 13 > 0 – 12
5x + 13 –13 > –12 – 13
Subtract 13 from both sides.
5x > −25
Divide both sides by 5 to isolate x.
x > −5

36. Example  Solve InEquality
Solve 8x + 13 > 3x – 12
SOLUTION Graph for x > −5 (
SOLUTION SetBuilder Notation
{x|x > −5}
SOLUTION Interval Notation
(−5, )

37. Example  AirCraft E.T.A.
An AirCraft is 150 miles along its path from Miami to Bermuda, cruising at 300 miles per hour, when it notifies the tower that The Twin-Turbo-Prop is now set on automatic pilot.
The entire trip is 1035 miles, and we want to determine how much time we should let pass before we become concerned that the plane has encountered Bermuda-Triangle trouble

38. Distance = [Speed]·[time]
Example  AirCraft E.T.A.
Familiarize
Recall the Speed Eqn:
Distance = [Speed]·[time]
So LET t ≡ time elapsed since plane on autopilot
Translate
300t = distance plane flown in t hours on AutoPilot
t = plane’s distance from Miami after t hours on AutoPilot

39. Example  AirCraft E.T.A. Translate the InEquality for Worry
Plane’s distance from Miami
Distance from Miami to Bermuda ≥

40. Example  AirCraft E.T.A. Carry Out
State: Since 2.95 is roughly 3 hours, the tower will suspect trouble if the plane has not arrived in about 3 hours

41. Example  CellPhone $Budget
You have just purchased a new cell phone. According to the terms of your agreement, you pay a flat fee of $6 per month, plus 4 cents per minute for calls.
If you want your total bill to be no more than $10 for the month, how many minutes can you use?

42. Example  CellPhone $Budget
Familiarize: Say we use the phone 35 min per month. Then the Expense
Now that we understand the calculation LET
x ≡ CellPhone usage in minutes per month

43. Example  CellPhone $Budget
Translate:
Or, With 0.04 = 4/100

44. Example  CellPhone $Budget
Carry Out

45. Example  CellPhone $Budget
Check: If the phone is used for 100 minutes, you will have a total bill of $6 + $0.04(100) or $10 
State: If you use no more than 100 minutes of cell phone time, your bill will be less than or equal to $10.

46. WhiteBoard Work Problems From §4.1 Exercise Set
62 (ppt), 53, 72, 80
Working Thru a Linear InEquality
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47. P4.1-62
Write InEquality for Passion greater-than, or equal-to Intimacy
Find Crossing Point
Thus Ans

48. Eric Heiden All Done for Today
Won Five Gold Medals and Set Five Olympic Records at the 1980 Winter Olympics

49. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –