Math 141 Chapter 1: Equations and Inequalities Notes Namestaff · Math 141, Jagodina Chapter 1 Notes Math lab may help 2.Example: Find an equation of the line through (-3,1) and (-4,3)

Math 141, Jagodina Chapter 1 Notes Math lab may help

Math 141

Chapter 1: Equations andInequalities

Notes

Name...............................................

1


1.1 The Coordinate Plane

Graphing Regions in the Coordinate Plane

1. Example: Sketch the region given by the set

(a) {(x, y)|x = −3}

(b) {(x, y)|y = 4}

2


(c) {(x, y)| − 2 < x ≤ 2}

(d) {(x, y)|x > 3 and y ≤ 3}

3


Distance Formula

2. Example: Consider the points (5, 0) and (0, 4).

(a) Plot the points in the coordinate plane.

(b) Find the distance between the points.

4


Midpoint Formula

3. Example: Find the midpoint of the segment that joins (5, 0) and (0, 8).

4. Example: If M(3,4) is the midpoint of the line segment AB and if A has coordinates(-1,2), find the coordinates of B.

5


Shifting the Coordinate Plane

5. Example: Suppose that each point in the coordinate plane is shifted 4 units to theright and 3 units downward.

(a) The point (3,2) is shifted to what new point?

(b) The point (a, b) is shifted to what new point?

Reflecting in the Coordinate Plane

6. Example: Suppose that the y-axis acts as a mirror that reflects each point to theright of it into a point to the left of it.

(a) The point (1,3) is reflected to what point?

(b) The point (a, b) is reflected to what point?

6


1.2 Graphs and Equations in TwoVariables

Graphing Equations by Plotting Points

1. Example: Sketch the graph of 3x− y = 5. Plot 5 points.

2. Example: Sketch the graph of y = x2 − 3. Plot 5 points.

7


3. Example: Sketch the graph of y = |x|. Plot 5 points.

Intercepts

4. Example: Find the x− and y−intercepts of y = x2 − 3.

5. Example: Find the x− and y−intercepts ofx2

16+

y2

25= 1 Graph the equation.

8


6. Example: Find the intercepts of the equation whose graph is shown.

−2 −1 1 2 3 4 5 6 7

−3

−2

−1

1

2

3

4

0

9


Circles Standard form of an equation of the circle with center (h, k) and radius r:

Standard form of an equation of the circle with center (0, 0) and radius r:

7. Example: Graph x2 + y2 = 9

8. Example: Graph (x− 2)2 + (y + 1)2 = 4

10


11


9. Example: Find an equation of the circle with radius 10 and center (1,-3)

10. Example: Show that the equation x2 + y2 − 2x + 6y − 15 = 0 represents a circle.Find the center and radius of the circle.

Symmetry

Symmetry with respect to the y-axis

Test: The equation must be unchanged when x is replaced with −x

Graph:

Symmetry with respect to the x-axis

Test: The equation must be unchanged when y is replaced with −y

Graph:

Symmetry with respect to the origin

Test: The equation must be unchanged when (x, y) is replaced with (−x,−y)

Graph:

12


11. Example: Test y = x5 − 9x3 for symmetry.

12. Example: Test y = x2 + 2x for symmetry.

13


1.3 Lines

Slope of a line

Point-Slope Form

Slope-Intercept Form

1. Example: Find the slope of each line.

(a) The line passes through (1,2) and (5,8)

(b) y = −3

2x + 4

(c) y = 4

(d) x = 5

14


2. Example: Find an equation of the line through (-3,1) and (-4,3).

Parallel Lines

Two nonvertical lines are parallel if and only if they have the same .

3. Example: Find an equation of the line through (2,5) that is parallel to the line6x + 4y + 5 = 0

Perpendicular Lines

Two lines with defined slopes are perpendicular if and only if their slopes are.

A vertical line is to a horizontal line.

4. Example: Find the equation of the line that is perpendicular to the line 6x+4y+5 =0 and passes through (1,1).

15


1.5 Modeling with Equations

Investement Problems

Interest=Principle × Interest Rate × Time (in years)I = Prt

1. Example: Melissa won $60,000 on a slot machine in Las Vegas. She invested partat 2% simple interest and the rest at 3%. In one year she earned a total of $1,600in interest. How much was invested at each rate?

Solution:Define a variable. Let x=

Organize relevant data into a table or a chart

principal × rate = interest1st account2nd account

Together

Equation:

16


Mixture Problems

2. Example: A chemist needs 180 ml of a 55% solution but has only 22% and 76%solutions available. Find how many ml of each should be mixed to get the desiredsolution.

Solve this problem by using only one variable.

Solution:

Define a variable. Let x=

Organize relevant data into a table.

Percent × ml = Amount of So-lution

Solution 1 (high %)Solution 2 (low %)Mixture (middle %)

Equation:

Solve!

17


Work Problems

The standard equation that will be needed for these problems is

Portion of job done in given time=work rate × time spent working

When two or more objects are working, the setup becomes

work rate of object 1 + work rate of object 2 =one job done× ×time spent working time spent working

or

time together

time apart+time together

time apart=one job done

3. Example: An office has two envelope stuffing machines. Machine A can stuff abatch of envelopes in 5 hours, while Machine B can stuff a batch of envelopes in 3hours. How long would it take the two machines working together to stuff a batchof envelopes?

Solution:

Define a variable. Let t

time together

time apart+time together

time apart=one job done

Equation:

18


4. Example: Mary can clean an office complex in 5 hours. Working together John andMary can clean the office complex in 3.5 hours. How long would it take John toclean the office complex by himself?

Solution:


Motion Problems

The standard formula that we’ll be using here is

Distance=Rate × Time

5. Example: Two cars are 500 miles apart and moving directly towards each other.One car is moving at a speed of 100 mph and the other is moving at 70 mph. As-suming that the cars start moving at the same time how long does it take for thetwo cars to meet?

Solution:

Draw a diagram.


Organize relevant data into a table.

19


Rate Time DistanceCar 1Car 2Total

Equation:

Solve!

20


1.6 Quadratic Equations

Solving using factoring

The zero factor property:If ab = 0 then either a = 0 and/or b = 0

1. Example: Solve by factoring

(a) 4m2 − 1 = 0 (b) 10z2 + 19z + 6 = 0 (c) 5x2 = 2x

21


Completing the Square

If you start with x2 + bx and add

(b

2

)2

, you’ll get a factorable trinomial:

x2 + bx +

(b

2

)2

factors as

(x +

b

2

)2

This process is called completing the square.

2. Example: Use completing the square to solve each equation.

(a) x2 + 6x + 1 = 0 (b) 2x2 + 7x + 2 = 0

22


Quadratic Formula

If ax2 + bx + c = 0, then

x =−b±

√b2 − 4ac

2a.

3. Example: Solve1

3x2 +

5

3x = 1 using the quadratic formula.

Applications of Quadratic Equations

4. Example: An object is thrown or fired straight upward at an initial speed of v0 ft/swill reach a height of h feet after t seconds, where h and t are related by the formula

h(t) = −16t2 + v0t.

Suppose that a bullet is shot straight upward with an initial speed of 800 ft/s.

(a) When does the bullet fall back to ground level?

(b) When does the bullet reach a height of 6,400 ft?

(c) When does it reach a height of 1 miles?

(d) How high is the highest point the bullet reaches?

23


1.7 Solving Other Types of Equations

Polynomial Equations

1. Example: Solve x3 = 16x

2. Example: Solve 5x3 − 5x2 − 10x = 0

3. Example: Solve 2x3 + x2 − 18x− 9 = 0

24


Rational equations

4. Example: Solve.10

x− 12

x− 3= −4

5. Example: Solve.

x

2x + 7− x + 1

x + 3= 1

25


Radical Equations

6. Example: Solve each equation.

(a)√

4− 6x = 2x

(b)√x + 1 + 2x = 8

26


Equations quadratic in form


(a) x4 − 5x2 + 4 = 0

Equations with Fractional Powers


(a) x43 = 4 (b) x

43 − 5x

23 + 6 = 0

27


1.8 Solving Inequalities

To solve an inequality, we must isolate the variable on one side of the inequalitysymbol. When both sides of an inequality are multiplied or divided by a negativenumber, the direction of the inequality symbol reverses.

1. Example: Solve each inequality. Give the solution set in set-builder notation andinterval notation.

(a) −3t + 11 < −1

(b)1

4z − 1

2≥ 2z

3+ 2

(c) 2 < 3x− 4 ≤ 9

28


Quadratic Inequalities

2. Example: Solve 2x2 − x > 1

3. Example: Solve (x− 3)2(x + 3)3 < 0

Rational Inequalities

4. Example: Solvex + 1

x + 4≥ 0

5. Example: Solve3 + x

3− x> 1

29


1.9 Absolute Value Equations andInequalities

Equations

Rule: |expression| = a⇒ expression=a or expression=−a where a ≥ 0

Before you can use the rule above, you must isolate the absolute value on the leftside.


(a) |x + 2| = 4

(b) 3|2x + 1|+ 2 = 8

(c) |x + 2| = |2x− 1|

30


Absolute value inequalities

Rule: |x| > c→ x > c OR x < −c

This rule applies to |x| ≥ c as well.

Rule: |x| < c→ −c < x < c

This rule applies to |x| ≤ c as well.

2. Example: Solve each inequality. Give the solution set in interval notation.

(a) |x + 1| > 4 (b) |2x− 3| ≤ 4 (c) |x + 1| ≥ −3

(d) |x + 3| < −2 (e) |2x + 1| ≤ 0

31

Documents

Math 141 Chapter 1: Equations and Inequalities Notes Namestaff · Math 141, Jagodina Chapter 1 Notes Math lab may help 2.Example: Find an equation of the line through (-3,1) and (-4,3)