A7_Topic 2: Equations and Inequalities Review Q1

A7_Topic 2

Equations and Inequalities Review

Understanding Word Problems- Understand the Relationship

A lot of word problems describe certain type of relationships.

For example:

Part + Part =Whole

Starting Amount + Addition =Total
Starting Amount -Deduction =Final Amount

Once you understand the relationship, you can substitute values to write an equation. 

Multiple Choice

The total cost of 10 doughnuts and 12 cookies at a bakery is $23.90. The cost of each cookie is $0.95.

• 10x = 22

• 10x + 12(0.95) = 23.90

• 10x =13.90 = 23.90

Example: The total cost of 10 doughnuts and 12 cookies at a bakery is $20. The cost of each cookie is $1.00.

• Cost of Doughnuts + Cost of Cookies =Total Cost

• 10d+12c=$20

• 10d+12(1)=20

Multiple Choice

New example: The total cost of 10 doughnuts and 12 cookies at a bakery is $23.90. The cost of each cookie is $0.95. Which equation represents the situation and how much is 1 doughnut?

• 10x = 23.90 x=$2.90

• 10x + 12(0.95) = 23.90 x=$1.25

• 10x =11.40= 23.90 x=$1.40

Understanding Word Problems- Understand the Relationship

You are making a model out of wood. You cut 2 pieces of wood that measure 4 1/3 inches each from a piece that is 24 inches long.

This is a part + part + part =whole relationship.

Two of the parts are exactly the same. So you can rewrite it as: 2(parts) + 1 part= whole.

Now that you understand the relationship, you can substitute values to write an equation. 

Multiple Choice

You are making a model out of wood. You cut 2 pieces of wood that measure 4 1/3 inches each from a piece that is 24 inches long. Which equation DOES NOT REPRESENT the situation?

•  $4\ \frac{1}{3}+2x\ =24$  

•  $2\left(4\ \frac{1}{3}\right)\ +\ x\ =24$  

•  $8\ \frac{2}{3}+x=\ 24$  

•  $24\ -\ 8\ \frac{2}{3}=\ x$  

Multiple Choice

You are making a model out of wood. You cut 2 pieces of wood that measure 4 1/3 inches each from a piece that is 24 inches long.  How long is the third piece of wood?

•  $8\ \frac{2}{3}$  

•  $15\ \frac{1}{3}$  
  

•  $16$  
  

Multiple Choice

While on a vacation you want to rent a bike. There is a initial $3 rental fee plus $1.50 for each hour. You can spend $24. How many hours can you rent the bike? Would you represent this with an equation or an inequality?

• Equation

• Inequality

• Neither

Multiple Choice

While on a vacation you want to rent a bike. There is a initial $3 rental fee plus $1.50 for each hour. You can spend $24.  How many hours can you rent the bike?

Which statment represents this relationship?

•  $3+1.50x=24$  

•  $3+1.50x\ >\ 24$  

•  $3+1.50x\ \le\ 24$  

Multiple Select

While on a vacation you want to rent a bike. There is a initial $3 rental fee plus $1.50 for each hour.  You can spend $24. How many hours can you rent the bike?

Select all that apply.

• 15

• 10

• 5

• 1

Understand the Relationship

The soccer team needs to raise at least $1,000 to attend a tournament The team raised $400 already and there are 3 months remaining until the tournament. Write an inequality statement that can be used to determine the dollar amount the team will need to raise during the remaining months?

This relationship is:
part + part >/= whole
starting amt. + #of months ($ amt. earned per month) >/=Total Raised

Multiple Choice

The soccer team needs to raise at least $1,000 to attend a tournament The team$400 and there are 3 months remaining until the tournament.

Would the team be ok with earning more than $1,000?

• Yes

• No

Multiple Choice

The soccer team needs to raise at least $1,000 to attend a tournament The team$400 and there are 3 months remaining until the tournament. 

Which inequality represents this situation?

•  $400+3x\ <\ 1,000$  

•  $400+3x\ \ge1,000$  

Multiple Choice

 $400+3x\ \ge1,000$  

Solve the inequality. Select the choice that describes the steps and provides the correct solution.

• First, undo subtraction by adding 400 to both sides. Then undo divsion by multiplying both sides by 3.  $x\ge500$  

•  $x\ge200$  First undo addition by SUBTRACTING 400 from each side. Then undo the multiplication by DIVIDING each side by 3.  

Using Distributive Property

• Sometimes you will need to use grouping symbols to represent a situation.

• When solving, you may have to distribute to simplify the expression.

Multiple Choice

A baker starts with 24 eggs. She makes 5 cakes. She has 9 eggs left over. Let x= the number of eggs used for each cake. Which equation would you use to find the number of eggs used per cake?

• 5x + 9 = 24

• 9x + 5 = 24

• 4(x + 9) = 24

Apply What You Know

• FInding the area of a trapezoid is a good example of solving an eqaution that invloves distributing.

•  $Area\ =\ \frac{1}{2}\left(base\ 1\ +\ base\ 2\right)height$  

• TIP: You can use Associative Property to rewrite and simplfy before distributing.

•  $\frac{1}{2}\left(h\right)\left(b1+b2\right)\ =Area\$  

• Multiply 1/2 and h before distributing.

Multiple Choice

 $\frac{1}{2}\left(4+b2\right)6=42$  

A trapezoid has one base of 4 inches, a height of 6 inches, and an area of 42 square inches. The equation below can be used to determine
b2, the length in inches of the second base of the trapezoid.

Select an equivalent equation.

•  $2+b2(6)=42$  

•  $12+3\left(b2\right)=42$  

More Formulas

• Area of a square: side*side or

   $A=s^2$  

• Area of a rectangle: length * width (height) or $A=lw$  

• Perimeter of a rectangle:  $2l+2w$  

• Perimeter of a square:  $4s$  

Multiple Choice

 $\frac{1}{2}\left(4+b\right)6=42$  

 $\frac{1}{2}\left(6\right)\left(4+b\right)=42$  

 $3\left(4+b\right)=42$  
 $12\ +3b=42$  

Find the meaure of b.

• b=18

• b=10

• b=6

Graphing Inequalities

• Inequalites represent a range of correct soultions so we can graph the solution as a ray on a numberline.

• When the circle is closed/ filled, then the solution includes that point.

• If the circle is open/ not filled, then the solution does not include that point.

Multiple Choice

 $x\le99$  Which graph represents the following inequality:

Multiple Choice

 $x\le131$  Which graph represents the following inequality:

Solving Inequalities

• First, simplfy by distributing and combining like terms as needed. Try to simplfy so that you have a 2-Step Inequality. Examples: Ax+B<C or A +BX>C

• Second, make a plan to isolate the variable to one side. Determine if you have to change the direction of the inequality symbol.

• Undo Addition or Subtraction by doing the opposite.

• Undo Multiplication or Division by doing the opposite.

• Check your inequality to make sure it makes sense.

Multiple Choice

 $8.5x\ \ge76.5$  

Solve the inequality.

• 9

• 68

• 85

Multiple Choice

 $10+\frac{x}{20}\le20$  

What is the first step to solve this inequality?

• Add 10 to both sides.

• Subtract 10 from both sides.

• Multiply  both sides by 10.

• Divide both sides by 10.

Multiple Choice

 $10+\frac{x}{20}\le20$  

What is the second step to solve this inequality?

• Add 10 to both sides.

• Subtract 10 from both sides.

• Multiply  both sides by 10.

• Divide both sides by 10.

To Flip the Symbol or Not?

• The inequality symbol will have to be flipped if when using inverse operations, both sides of the inequality are multiplied or divided by a negative.

• Tip: If the variable term is negative, then you will have to divide or multiply both sides by a negative.

• Remember to capture the negative sign; subtracting is the same as adding the opposite.

Multiple Choice

 $-2x\ +\ 20\ \ge100$  

Do you need to flip the sign after you solve so that you have a true statement?

• Yes

• No

Multiple Choice

 $400\ +\ 4n\ \ge750$  

Do you need to flip the sign after you solve so that you have a true statement?

• Yes

• No

Multiple Choice

 $15\ +\ -\frac{x}{5}\ <45$  

Do you need to flip the sign after you solve so that you have a true statement?

• Yes

• No