Name:Period:

Algebra

Unit 2: Graphing and Solving Equations and Inequalities

Day 1-2 Solving InequalitiesDay 3 Solving Inequalities with Largest/Smallest Integer ValueDay 4 Graphing Lines without a CalculatorDay 5 Graphing Lines with a CalculatorDay 6-7 Given a Line Write the EquationDay 8 Given a Table Write the Equation/QUIZDay 9-11 Graphing InequalitiesDay 12 Graphing Systems of Inequalities (Dividing by Negatives)Day 13-14 Graphing Systems of Inequalities (Word Problems)Day 15 Given a Graph Write the InequalityDay 16 ReviewDay 17 Test

Day 1: Solving Inequalities

Notice that unlike equations, inequalities tend to have ______________ values that make them true.

We can determine the ___________________ of an inequality using values for the variables.

Less than Less than or equal to Equal Greater than or equal to Greater than

Exercise 1: For each inequality, state whether it is true or false.

(1) 10<17 (2) 9≤−6 (3) 0>−4 (4) 14≥14

We can also test if inequalities are _______________ for different values.

Exercise 2: Given the inequality 3 ( x−2 )≥2 x+1determine if it is true or false for the following values of x.

(5) x=10 (6) x=1 (7) x=7

Exercise 3: For each of the following inequalities, determine if it is true or false at the given value of the variable.

(8) (9)

Solve the following inequalities and graph the solution set on a number line. Give the solution in interval or set notation.

(1) 4 x−3≥5 (2) 8−2x>16

(3) ( x−2 )−3 (2 x+1 )≤7 x+4−3(x+1) (4)

Day 2: Solving Inequalities

Solving inequalities is just like solving equations except when you _______________________ or _______________________ by a negative number, you must FLIP the inequality!!!!!

Solve the following inequalities and graph the solution set on a number line. Give the solution in interval or set notation.

(1) (2)

(3) (4)

(5) (6)

(7) (8)

Just as we can solve many real-world problems involving linear equations, there are plenty of situations whenan _______________________ is called for instead.

(9) Keith has $500 in a savings account at the beginning of the summer. He wants to have at least $200 in the account by the end of the summer. He withdraws $25 each week for food, clothes and movie tickets. Write an inequality that represents Keith’s situation.

How many weeks can Keith withdraw money from his account? Justify your answer.

(10) A fish tank can hold at most 315 gallons of water. If a hose is filling the fish tank at a rate of 15 gallons every 10 minutes, how many hours can the hose be left on before the tank overflows?

Day 3: Solving Inequalities with Largest/Smallest Integer Value

Sometimes, the most difficult part of answering a question about inequalities is the wording. You will sometimes be asked for the smallest or largest _______________________ in the solution set.

For example, say you solved an inequality to find your solution to be the following:

3

The question is asking for “the smallest integer in the solution set.” What is your answer? Why?

Given the following solution sets, give the specified integer.

(1) Largest integer:

10

(2) Largest integer:

12

(3) Smallest integer: -13

(4) Solve the inequality below to determine and state the smallest possible integer value for x in the solution set.

x−3<5 x−3

(5) Given 2 x+ax−7>−12, determine the largest integer value of a when x=−1.

(6)

(7) Determine the largest integer that makes 4 x+12≤15true.

Day 4: Graphing Lines without a Calculator

The chances that you will be stranded on a desert island without a graphing calculator and asked to graph a linear equation are fairly slim. HOWEVER, graphing linear equations without your calculator can be much faster and will allow you to know how a line will look simply by looking at its equation.

Slope-Intercept Form

y=mx+b

m=¿b=¿

Graphing a linear equation:1. Identify m and b2. Plot b on the y-axis3. From b, count out your slope to plot each point4. Connect the points, add arrows and label the line

It is a good idea to do a visual check to make sure you graphed your slope properly.

Negative Slope Positive Slope

Exercise 1: Graph the following linear equations.(1) y=3 x+1 (2) f (x)=−2x+4

Slope is best thought of as a fraction!

_________ or _________

Sometimes, you will be given a linear equation that is not in ________________________________________ form. Before you can successfully identify m and b, it is best to put the equation in slope-intercept form. This is done by solving for ____.

Graph the following linear equations.

(1) 5 y=3 x−10 (2) f (x)+4 x−1=5+3 x

(3) 17−2x=3x−1−2 y

Day 5: Graphing Lines with a Calculator

Graphing linear equations with your calculator is a great way to check that you have plotted your points correctly.

1. Get the equation in ___________________________ form.2. Go to your y=¿ screen and type the equation.3. Press 2nd>graph to see your table.4. Copy your table down on your paper. Only include integer points that fit on your graph.

Exercise 1: Use your calculator to graph the following linear equations.

(1) f (x)=2(3 x−4)

(2) 3 f (x)+4 x=−21

Exercise 2: Look back at the table you created for #1. What pattern do you see in the y column? What does that pattern correspond to?

Exercise 3: Carlesha recorded the following table from her calculator. What is the slope of the following linear equation that she is graphing? Justify your answer.

x y-1 20 51 82 113 14

Exercise 4: Arrange the following linear equations from least steep to most steep slopes. (Use your calculator to help you!)

y=3 x y=12x y=20x y= 9

10x y=x y=5 x

What pattern do you notice? How do you think this pattern transfers to negative slopes? Explain.

Day 6-7: Given a Line Write the Equation

You can use the knowledge that you have of graphing linear equations to move backwards!If you are given a graph of a linear equation, you can write the equation.

To write a linear equation, you only need two pieces of information from the line: __________ and __________________

Exercise 1: Read the slope and y-intercept from the following graphs.

(1)

(2)

To Find the Slope: Identify two points that fall perfectly on

points with integer coordinatesLook for this: Not this:

Count how many units you move _____ Count how many units you move _________ Decide if your slope is _________________ or

_________________

To Find the Y-intercept: Find where the line crosses the ____-axis

o This is where x=¿¿ Take the y coordinate of this point

Once you read these from the graph, substitute them into the slope-intercept equation. y=∎ x+∎Voilà, you’ve written the equation!

Write the equation of each line you see graphed.

(1)

(2)

The scale will not always be one box=one unit. When the scale is different, you must calculate slope accordingly.

Exercise 1: Read the slope from the following graphs.

(1)

2)

(6 ,−14)

Write the equation of each line you see graphed.

(1)

(2)

(−33,10)

Day 8: Given a Table Write the Equation

Just like we can use a graph to write the equation of a line, we can also write the equation given a table.

To write a linear equation, you only need two pieces of information from the line: _____ and _____

Exercise 1: Read the slope and y-intercept from the following tables.(1)

(2)

Exercise 2: Sometimes the x column will not be increasing by 1. You can still find the slope.

(3) (4)

To Find the Slope: Look in the ____ column of your table The values should be ________________ (+m)

or ______________________ (−m) at a constant rate

_______________= ______________________

This rate is your slope!

To Find the Y-intercept: Find where x=¿¿ Take the y coordinate of this point

x y-3 0-2 4-1 80 121 162 20

x y0 81 62 43 24 0

x y-2 -100 02 104 206 30

x y-5 240 185 12

10 615 0

Write the equation of each line you see graphed.

(1)

(2)

(3)

(4)

(5)

x y-1 30 01 -32 -63 -9

x y-4 3-3 3.5-2 4-1 4.50 5

x Y-4 -60 -74 -88 -9

12 -10

x y1 132 163 194 225 25

x y4 78 9

12 1116 1320 15

Day 9-11: Graphing Inequalities

When graphing linear inequalities, you begin with the same process as when you graph an _____________________. Summarize the process below:

Then, things get a little tricky!

You must decide if your line is solid or dotted.

When you graph a line, a solution is ________________________________________________.When you graph a linear inequality, you must _______________ a region of your graph. Every point in this region is also a ___________________________________.

You must decide where to shade. The best way is to select a __________________________ that is NOT on the line!

If it is FALSE, shade ____________________________ your test point.If it is TRUE, shade _____________________________ your test point.

Exercise 1: Determine where to shade for the solution set of each inequality.

(1) y ≥12x−3 (2) y← x−1

Inequality In the solution set?Solid Line

Dotted Line

(1) y ≤−4 x+1

(2) 3y>2 x−6

(3) y−5←x+1

(4) y−6 x ≥1

(5) y>1

(6) 5 y+15<10 x

(7) x≥−4

(8) 2 y−4≥10

(9) −5 x<5 y−5

(10) 3 y+9−21 x+6≤0

(11) −5 x<15 y−30

(12) y−4←45x

(13) y>x

(14) y ≥7 (x+1)

(15) y−1<2 ( x−4 )+3

(16) 4 y←7x

(17) y+27x−6≤ 6

7x−4

(18) 2( y+1)←8 x−6

(19) y+73x ≤ 23x

(20) y+4<35(x−5)

(21) 2( y−5)≥3(x−4)

Day 12: Graphing Systems of Inequalities

When you solve a system of inequalities graphically, you’re looking for where the solution sets _________________________. Plot each inequality, mark the overlap in the solution sets with a big S, for __________________________.

Remember, a point on a ____________ is in the solution set. A point on a _______________________ line is NOT in the solution set.

y< 12x+2

y> 12x+5

Graph the system of inequalities and give a point in the solution set.

Day 13-14: Graphing Systems of Inequalities (Word Problems)

Systems of linear inequalities model many situations in real life. You organize the given information into two inequalities that must be true.

(1) Mary babysits for $10 per hour. She also works as a tutor for $5 per hour. She is only allowed to work 13 hours per week. She wants to make at least $70. Write and graph a system of inequalities to represent this situation.

(2) A salad contains ham and chicken. There are at most 6 pounds of ham and chicken in the salad. Write and graph a system of inequalities to represent this situation. Chicken is $2 per pound and ham is $3 per pound. You can spend no more than $18.

(3) At the convenience store, milk costs $2 per gallon and almond milk costs $3 per gallon. You have at most $18 to spend on milk. Write and graph a system of linear inequalities to represent this situation. You cannot carry more than 5 items.

(4) You can work a total of no more then 41 hours each week at your two jobs. Housecleaning pays $5 per hour and your sales job pays $8 per hour. You need to earn at least $254 each week to pay your bills. Write a system of inequalities that shows the various numbers of hours you can work at each job.

(5) A clothing manufacturer has 1000 yd. of cotton to make shirts and pajamas. A shirt requires 1 yd. of fabricand a pair of pajamas requires 2 yd. of fabric. It takes 2 hr. to make a shirt and 3 hr. to make the pajamas,and there are 1600 hr. available to make the clothing.

(6) A potter is making cups and plates. It takes her 6 min. to make a cup and 3 min. to make a plate. Each cup uses 3 lb. of clay and each plate uses 1 lb. of clay. She has 20 hr. available to make the cups and plates and has 250 lb. of clay.

Day 15: Given a Graph Write the Inequality

Writing an inequality from a graph is very similar to when we wrote ______________________ from graphs.You must determine ______________ and_____________ and substitute them into _____________________________.

Then, instead of an = sign, you must determine which inequality to use.

If you see a ____________________ line use _____ or ______. If you see a ____________________ line use ______ or _______.

Decide which way to face the inequality based on solution set _______________________________.For a solution set above the line use _________ or _________.For a solution set below the line use _________ or _________.

1. Select a point.2. Determine if it is shaded on your graph3. __________________ the point into your inequality (without an inequality—just leave a blank for now)4. Face the inequality in the appropriate direction.

Shaded=__________________ Not shaded=_________________

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Blanks

Day 1EquivalentTrueTruth valueManyExpressionsMultiplyDivide

Day 2inequality

Day 3Integer

Day 4Rise over run Change in y over change in xSlope-interceptY

Day 6M, bUp, overPositive, negative

Day 8M and bYIncreasing, decreasingRise over run, change in y over change in x0

Day 9-11Linear equation/lineAny point on the lineShadeSolutionTest pointAway fromTowardsDay 12

SolutionSoliddotted

Day 15EquationsM, bY=mx+bDottedsolidtest pointSubstituteTruefalse