4 3 compound inequalities

Today in class we were introduced to compound inequalities.

One thing we learned about compound inequalities is the meaning of it. Compound means more than one. This is why it makes sense that compound inequalities have more than one restriction.

An example of types of compound inequalities are when both restrictions apply to the inequality or when one or the other restriction applies to the inequality. The type that has both restrictions apply is one that uses the word "and". "and" in that type of compound inequality tells us that both parts of the inequality must be true. The type of compound inequality that one or the other restriction applies to the inequality is one that uses the word "or". "or" in that type of compound inequality tells us that either restriction can be used. An example of a compound inequality that uses "and" is x>-5 and 7>x. It also can be written as a combined inequality as -5. An example of a compound inequality that uses "or" is x>-12 or x<8.

You graph these with filled or blank circles just like the way you do with other inequalities. The only difference is that when you graph compound inequalities that use "or" you skip space that has numbers that are not part of the problem and it tells you the variable can't be any of those numbers. The graphs for compound inequalities that use the word "and" limit the numbers that the variable could be to the numbers that are connected by circles.

You solve compound inequalities with the same rules and the same way you solve regular inequalities.

Compound Inequalities

Here are the notes on Compound Inequalities.

4 3 Compound Inequalitites

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Peter's Formulas

Take a look at Peter's website on gravitation and escape velocity of planets & stars. Click on the title to go to his presentation.

Bridget's Formulas

Take a look at the formulas that Bridget examined: Average Yards and Grade Point Average. Click on the title to see her presentation.

Harry's Formulas

Take a look at Harry's formulas: On Base Percentage and body mass index. Click on the title to view his presentation.

Mike's Formula

Take a look at Mike's formulas on medication dosage and hockey team rankings. Just click on the title to see his presentation.

Kathryn's Formulas

See how Kathryn's dad uses formulas in his job as a printer. Click on the title to view her presentation.

Ryan's Formulas

Check out Ryan's formulas for calculating dosage and cancer risk. Click on the title to view his presentation.

Joe's Formulas

Check out Joe's formulas on WHIP and temperature conversions. Please click on the title to view his presentation.

Noah"s Formulas

Noah presents the formula for airplane lift and the "magic number" for baseball. Take a look by clicking on the title.

Diana's Formulas

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Alex's Formulas

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Allie's Formulas

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Emily's Formulas

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Nick's Formulas

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Kate's Formulas

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Ada's Formula Project

Take a look at Ada's presentation on the use of formulas in electrical engineering. Take a minute to complete the rubric and give her some feedback. Just click on the title to view her presentation.

Formula Scoring Rubric

Inequalities

Today in class we got introduced to the solving and graphing of inequalities.

Example 1:
You must be at least 35 inches tall in order to ride Pirate's Lagoon.
Since "at least 35 inches" includes everyone who is exactly 35 inches, the inequality should show that x equals or is greater than 35.

Graphing Inequalities: Inequalites can be graphed on number lines. A filled-in circle means that the number that it is over is included in the set of numbers, and a blank circle means the number isn't included. For example, using the previous example, 35 would have a circle over it, and was filled in. If instead the situation read that you must be taller than 35 inches, exactly 35 inches isn't included. Therefore, the circle above it on a number line would not be shaded in.

Solving Inequalites: Inequalites can be solved the same way as equations, which we learned earlier in the year. Only instead of an equal sign, you have a greater than, less than, equal to or greater than, or equal to or less than.

IMPORTANT!!!!!!!!!!!!!!!!!!!!
When you multiply or divide an inequality by a negative number, you must reverse the direction of the inequality symbol.

5>4 true
-5>-4 false
-5<-4 true


Triangle Inequalities: Today we also got to play around with triangle inequalities. We each drew 3 triangles (any size or type) and collected data. We found the lengths of the shortest, medium, and largest sides. We put our information into a table.

We realized that in order to be a triangle, the shortest side and the middle side together had to be larger than the longest side.

Inequalities

Notes and examples on graphing and solving inequalities.

Inequalities

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Moving Straight Ahead

Here are the notes from today.

Moving Straight Ahead 4.4

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Today we continued to work on slope and the y-intercept. We did problem 4.4 which showed how you can tie those two concepts into every day life. Part A was about a girl who got money on her birth day but wouldn't tell her sister how mush so instead she gave her clues. The first clue was after 5 weeks I will have $175 and the second clue was after eight weeks she will have $190. She saves a certain amount of her allowance each week. when it asks for the amount of her allowance she saves per week she it is actually asking for the slope or rate of change. When it asks for the amount of money she receives from her grandfather you are finding the y-intercept. This is the equation you would use.

A= total amount of $ saved

W= # of weeks

A=5w + 150

The 5 is the slope but it is also the part of her allowance she saves each week. The 150 is the y-intercept or where she starts but is also the amount her grandfather gives her.

Nick S.

Finding Slopes

In class on Monday we learned about finding slopes on tables, graphs, or equations.

The first thing we did was find the slope of the two staircase measurements we took at home. First we put the rise over the run in fraction form and then we divided the rise by the run. The standard ratio is between 0.45 and 0.60 but most peoples ratios were greater. For example if the rise of a stair case was 6 inches and the run was 7 inches we would put that into the fraction 6/7 and into the ratio .86 this would not be within standards.

We then took notes on the slope of something learning that the slope is how much the y-axis increases or decreases or stays the same when the x-axis goes up by one. In the equation 6x+10=y the slope is 6. The slope is also called the rate of change which we have learned about previously.

We then learned that if the y-axis gets higher as the x-axis does it has a positive slope and if the y-axis gets lower as the x-axis increases it has a positive slope. Some examples of a positive slope are 3x+5=y, x-1=y, and 5x=y. Some examples of negative slopes are 6-9x=y, -7x=y, and -x+6=y.

Some examples of all of this are 9x+10=y, which has a positive slope and a slope of 9, -9x-8=y, which has a negative slope and a slope of -9, and 8=y which has no slope and a slope of 0.

Moving Straight Ahead - slope of a line

Notes from today on slope of a line from an equation, table and graph.

Moving Straight Ahead 4.2

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Problem 3.5: Finding the Point of Intersection

In class on Friday, we continued the problem on finding the point of intersection.

The point of intersection is where two sets of data break even. In a table, the two columns are the same at the same value of x, and on a graph, the two sets break even when the two lines intersect., and in an equation you make each equation equal the other and then solve for the variable. In the problem with Fabian's Fabulous Bakery, the equation for his expenses was E=825+3.25n, and the equation for his income was I=8.20n. In this case, the point of intersection is where Fabian no longer owes any money and is ready to start making a profit. To use the equation, you would put the expression for his expenses on one side and the expression for his income on the other side, and then solve for the variable. The equation to find the break even point for Fabian would be 825+3.25n=8.20n. The solution is 166 2/3, so Fabian would have to sell 167 cakes to be out of debt because he cannot sell 2/3 of a cake.

Post by: Peter S.

Y-Intercepts and Coefficients

Today we continued the work we did yesturday on y-intercepts and coefficints.

To review, the y-intercept is the point were the line crosses the y axis, and the coefficient is the number multiplied by the variable. In the Emile and Henri problem, Henri's equation is d=1t+45. 1 is the coefficent of t, and the y-intersect is 45.

Using this we started Problem 3.5 on page 55. This problem was about Finding the Point of Intersection.

"At Fabulous Fabians's Bakery, the expenses E to make n cakes per month is given by the equation E=825 +3.25n. "
In this equation, we are identifying things like the y-intercept (825), and th coefficient (3.25) and finding the income using another formula I=8.2n.

Allie G.

Problem 2.3 Comparing Costs

Today we learned about the y-intercept. Here are some useful definitions:

y-intercept: the point where the line crosses the y-axis on a graph or when x=0 in a table (the starting point)

coefficient: the rate of change AND/OR the number that multiplies a variable in an equation

In the equation y=mx+b, y=the dependant variable, m=the coefficient and/or rate of change, x= the independent variable, and b=the y-intercept.

Here's another example, using the problem with Emile and Henri that we did yesterday in class.

Henri's equation: d=1t+45
The coefficient of t is 1 and the y-intercept is 45

Emile's equation: d=2.5t
The coefficient of t is 2.5 and the y-intercept is 0.

If no y-intercept is stated in the equation (that means there is no addition), it is assumed to be 0.

Moving Straight Ahead problem 2.3 - y=mx + b

Notes from today's lesson on the y = mx + b form of linear equations.

Moving Straight Ahead 2.3

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Problem 2.1 Finding the Point of Intersection

Today in class we learned how to find the variable so that the two equations are equal. This is the problem that we focused on.

In Ms. Chang's class, Emile found out that his walking rate is 2.5 meters per second. When he gets home from school, he times his little brother Henri as Henri walks 100 meters. He figured out that Henri's walking rate is 1 meter per second.
Henri challenges Emile to a walking race. Because Emile's walking rate is faster, Emile gives Henri a 45-meter head start. Emile knows his brother would enjoy winning the race, but he does not want to make the race so short that it is obvious his brother will win. How long should the race be so that Henri wins a close race?

One option is to use a graph. First, you plot the points to represent how far they have walked after every ten meters and connect them. Then you look to see where the two lines intersect to find the distance they will have to walk to meet. If you subtract a meter from that, Henri will only win by a little. The lines on the graph intersect at 75 meters so if you subtract one and make it 74, Henri will win a close race.

Another option is to use a table. This is what it would look like:

Time___Emile __Henri
_ 0 ____ 0______45
_10____ 25_____ 55
_20____ 50_____65
_30____ 75_____75
_40____100____ 85

(The time is in seconds and the data under Emile and Henri represent how far they have walked in meters.)

After 30 seconds they meet at 75 meters. If you make the race 74 meters, Henri will win but just by a little.

A third option to solve this is to use an equation. That is what I did.

2.5x = 1x + 45
2.5x - 1x = 1x + 45 - 1x
1.5x = 45
1.5x / 1.5 = 45 / 1.5
x = 30

2.5 = Emile's walking rate
x = seconds
1 = Henri's walking rate
45 = Henri's starting point

After 30 seconds they will meet. To figure out how far they will be from the starting point you have to do the check:

2.5(30) = 1(30)+45
75 = 30+45
75 = 75

After you do the check you figure out that at 75 meters they will meet. You need to subtract a meter so that Henri will win. After they have walked 74 meters, Henri will only be ahead by a little so he would win a close race.

From the graph, table and equation you can figure out that they will meet after 30 seconds at 75 meters. For Henri to win a close race, it would have to be 74 meters long.

Post by: Alex S.

Moving Straight Ahead

Lesson on linear functions using tables, graphs & equations.

Moving Straight Ahead 2.1 & 2.2

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Moving Straight Ahead

Notes on Moving Straight Ahead - linear relationships.

Moving Straight Ahead

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