[email protected] MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.

[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§4.1 Solve§4.1 Solve InEqualities InEqualities



Review §Review §

Any QUESTIONS About• §2.5 → Point-Slope Line Equation

Any QUESTIONS About HomeWork• §2.5 → HW-7

2.5 MTH 55



Solving InEqualitiesSolving InEqualities

An inequality is any sentence containing

, , , , or . Some

Examples3 2 7, 7, and 4 6 3.x c x

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.



Linear InEqualitiesLinear InEqualities

A linear inequality in one variable is an inequality that is equivalent to one of the forms that are similar to mx + b

0or0 baxbax

where a and b represent real numbers and a ≠ 0.

0or0 baxbax



Example Example Chk InEqual Soln Chk InEqual Soln

Determine whether 5 is a solution to

a) 3x + 2 >7 b) 7x − 31 ≠ 4

SOLUTIONa) Substitute 5 for x to get 3(5) + 2 > 7, or

17 >7, a true statement. Thus, 5 is a solution to InEquality-a

b) Substitute to get 7(5) − 31 ≠ 4, or 4≠ 4, a false statement. Thus, 5 is not a solution to InEquality-b



““Dot” Graphs of InEqualitiesDot” 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



Example Example Graph InEqualities 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

6-3 -1 1 3 5-4 0 4-2-4 2 6-3 -1 1 3 5-4 0 4-2-4 2




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

3-6 -4 -2 0 2-7 -3 1-5-7 -1 3-6 -4 -2 0 2-7 -3 1-5-7 -1 3-6 -4 -2 0 2-7 -3 1-5-7 -1




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.”

5-4 -2 0 2 4-5 -1 3-3-5 1 5-4 -2 0 2 4-5 -1 3-3-5 1 5-4 -2 0 2 4-5 -1 3-3-5 1

• Note the– OPEN dot at −3 → due to −3< x

– CLOSED dot at 5 → due to x≤5



Interval NotationInterval 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



Interval vs Dot NotationInterval vs Dot Notation Graph x ≥ 5

[

Dot Graph

Interval Graph

Graph x < 2

)

Dot Graph

Interval Graph



Interval Graphing of InEqualitiesInterval 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.



Set Builder NotationSet Builder Notation

In MTH55 the INTERVAL form is preferred for Graphing InEqualities

A more compact alternative to InEquality Solution Graphing is SET BUILDER notation:

3| xxRead as: “x such that x is…

SET BUILDERNotation



Compact Interval NotationCompact 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, )



Example Example SetBuilder & Interval 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 ]



Example Example Set Builder 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 (



Intervals on the Real No. LineIntervals on the Real No. Line



Addition Principle for InEqsAddition 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 Example Addition Principle Addition Principle

Solve & Graph 26 x

Solve (get x by itself)6266 x

4xAddition Principle

Simplify to Show Solution

• Any number greater than −4 makes the statement true.

Graph(



Multiplication Principle for InEqsMultiplication 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 ≥



Mult. Principle SummarizedMult. Principle Summarized

Multiplying both Sides of an Multiplying both Sides of an Inequality by a NEGATIVE Inequality by a NEGATIVE Number REVERSES the Number REVERSES the DIRECTION of the InequalityDIRECTION of the Inequality• Examples

61833263 xx

101411014 xxxx



Example Example Solve & Graph Solve & Graph

Solve & Grapha) b)

Soln-a)

204 y 47

1x

Divide Both Sides by −4

Graph

4

20

4

4

y

Reverse Inequality as the Eqn-Divisor is NEGATIVE5y

The Solution Set: {y|y > −5}

(




Soln-b) 47

1x

Multiply Both Sides by 7

Graph

747

17 x

28x Simplify

The Solution Set: {x|x ≤ 28}

3015 255 2010 303015 255 2010 303015 255 2010 30 3015 255 2010 303015 255 2010 303015 255 2010 30]



Example Example Add & Mult Principles Add & Mult Principles

Solve & Graph 1014 xx SOLUTION

110114 xx

94 xx

Add ONE to Both sides

Simplify

94 xxxx Subtract x from Both Sides

933

1x Divide Both Sides by 3

3x Simplify & Show Solution

]



Example Example Solve Solve 33xx −− 3 > 3 > xx + 7 + 7

Soln Add 3 to Both Sides

Simplify

The Solution Set: {x|x > 5}

37333 xx103 xx

103 xxxx Subtract x from Both Sides

2

102 xDivide Both Sides by 2

5x Simplify

Graph

8-1 1 3 5 7-2 0 4 82-2 6 8-1 1 3 5 7-2 0 4 82-2 6 8-1 1 3 5 7-2 0 4 82-2 6(



Example Example Solve Solve 15.4 15.4 −− 3.2 3.2xx < < −−6.766.76

Soln To Clear Decimals

Dist. in the 100

The Solution Set: {x|x > 6.925}

76.62.34.15100 x

76.61002.31004.15100 x

6763201540 x Simplify

154067632015401540 x Subtract 1540

2216320320

1

x Simplify; Mult. By −1/320

925.6320

2216

x Simplify; note that Inequality REVERSED by Neg. Mult.




Solve 934735 xxx

Soln

91247155 xxxUse Distributive Law to Clear Parentheses

34152 xx Simplify

3343152 xx Add 3 to Both Sides

xx 4122 Simplify

xxxx 421222 Add 2x to Both Sides

6

1612 x Simplify; Divide

Both Sides by 6




Solve 934735 xxx

Soln xx

2

6

1612 From Last Slide

Graph

The Solution Set: {x|x ≤ –2}.

2x Put x on R.H.S.; Note Reversed Inequality

2-7 -5 -3 -1 1-8 -6 -2 2-4-8 0 2-7 -5 -3 -1 1-8 -6 -2 2-4-8 0 2-7 -5 -3 -1 1-8 -6 -2 2-4-8 0]



Equation ↔ InequalityEquation ↔ Inequality

EquationReplace

= byInequality

x = 5 < x < 5

3x + 2 = 14 ≤ 3x + 2 ≤ 14

5x + 7 = 3x + 23 > 5x + 7 > 3x + 23

x2 = 0 ≥ x2 ≥ 0



Terms of the (InEquality) TradeTerms 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.



Terms of the (InEquality) TradeTerms 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

x < 5, or (–∞, 5))5



Terms of the (InEquality) TradeTerms 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.



THE NON-NEGATIVE IDENTITYTHE NON-NEGATIVE IDENTITY

for ANY real number x

x2 0

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



Solving Linear InEqualitiesSolving Linear InEqualities1. Simplify both sides of the inequality as needed.

a. Distribute to clear parentheses.b. Clear fractions or decimals by multiplying through

by the LCD just as was done for equations. (This step is optional.)

c. Combine like terms.

2. 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.

3. 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.



Example Example Solve InEquality Solve InEquality

Solve 8x + 13 > 3x − 12 SOLUTION

8x − 3x + 13 > 3x − 3x − 12

Subtract 3x from both sides.

Subtract 13 from both sides.

Divide both sides by 5 to isolate x.

5x + 13 > 0 – 12

5x + 13 –13 > –12 – 13

5x > −25

x > −55

25

5

5 x



Example Example Solve InEquality Solve InEquality

Solve 8x + 13 > 3x – 12 SOLUTION Graph for x > −5

SOLUTION SetBuilder Notation

{x|x > −5} SOLUTION Interval Notation

(−5, )

(



Example Example AirCraft E.T.A. 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



Example Example AirCraft E.T.A. 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

• 150 + 300t = plane’s distance from Miami after t hours on AutoPilot




Translate the InEquality for Worry

Plane’s distance from Miami

Distance from Miami to Bermuda

≥

150 300t 1035

150 300t 150 1035 150




Carry Out 300t 885

300t

300

885

300t 2.95

State: Since 2.95 is roughly 3 hours, the tower will suspect trouble if the plane has not arrived in about 3 hours



Example Example CellPhone $Budget 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?




Familiarize: Say we use the phone 35 min per month. Then the Expense

month

47

month

min 35

min

040

month

6 .$.$$

Now that we understand the calculation LET• x ≡ CellPhone usage in minutes per month




Translate:

$10Than Less

Be To

Expense

MinutePlus

Expense

Montly

$100.04$6 x Or, With

0.04 = 4/10010

100

46 x




CarryOut




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.



WhiteBoard WorkWhiteBoard Work

Problems From §4.1 Exercise Set• 62 (ppt), 53, 72, 80

Working Thrua LinearInEquality



P4.1-62P4.1-62 Write InEquality for Passion greater-than, or equal-to Intimacy

Find Crossing Point

Thus Ans

}{

),[

5

or 50

xx



All Done for TodayAll Done for Today

EricHeiden

Won Five Gold Medals and Set Five Olympic Records at the 1980 Winter Olympics



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

[email protected] MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.

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