Integers, equations and inequalities

Integer Operations with Word Problems

Adding Integers

• SAME SIGNS: Keep the same sign in your answer and ADD

• Ex: 3 + 4 =7

• Ex: (-3) + (-4) = -7

• DIFFERENT SIGNS: Keep the sign of the number with the larger magnitude and SUBTRACT

• Ex: 7 + (-3) = 4

• Ex: (-7) + 3 = -4

Multiple Choice

5 + 4

• 20

• 1

• 9

Multiple Choice

(-9) + 2

• -11

• -7

• -18

• 11

Multiple Choice

5 + (-5)

• 0

• 10

• 1

• -10

Subtracting Integers

• Rewrite any subtraction problem using the rule: "Keep, change, change"

• Ex: 3 - 5 is rewritten as 3 + (-5)

• Then use the addition rule that applies.

• Ex: 3 - 5 becomes 3 + (-5) = -2

Multiple Choice

Rewrite 9 - (-6) as an addition problem

• 9 + 6

• -9 + 6

• 9 + (-6)

• 9 - (-6)

Multiple Choice

3 - (-7)

• -4

• -10

• 4

• 10

Multiple Choice

-18 - 7

• -25

• -11

• 11

• 25

Multiplying and Dividing Integers

Multiple Choice

$$2\div2$$  

• 1

• 0

• 4

Multiple Choice

$$\left(-9\right)\times\left(-3\right)$$  

• -11

• -6

• -27

• 27

Multiple Choice

$$\left(-11\right)\times3$$  

• -8

• 8

• -33

• 33

Multiple Choice

$$-12\div6$$  

• 2

• -2

• 6

• -6

Multiple Choice

$$\left(-30\right)\div\left(-6\right)$$  

• -36

• 36

• -5

• 5

Poll

How do you feel about integer operations?

• I still need help

• I could use a little more practice

• I get it

• I'm ready for a challenge

Multiple Choice

The temperature on Sunday was -15^o C. The next day, the temperature went down 12 degrees. What was the temperature the next day?

• -27

• 27

• -3

• 3

Multiple Choice

A submarine was 52 ft. below sea level. It ascended 38 ft. toward the surface. What is the depth of the submarine?

• 90 ft below sea level

• 26 ft below sea level

• 14 ft below sea level

Multiple Choice

On a certain day, the temperature changed at a rate of -2 degrees per hour. How long did it take for the total change in temperature to be -14 degrees?

• 7 hours

• 5 hours

• -7 hours

• -5 hours

Multiple Choice

Lindsay owes 3 people $5 each. Which value shows her total debt?

• $15

• $2

• -$15

• -$8

Solving two-step equations

• SWBAT use inverse operations to solve multi-step equations.

• SWBAT develop strategies to solve multi-step equations.

• NC.7.EE.3 Solve multi-step real-world and mathematical problems posed with rational numbers in algebraic expressions.

• NC.7.EE.4 Use variables to represent quantities to solve real-world or mathematical problems.

Standards and objectives

Steps for solving a Two Step Equation

​!Goal: Work to isolate the variable!

1. Use an inverse operation to cancel out the term not connected with the variable. (Do this first)

2. The use an inverse operation to get the variable by itself

3. (Recommended) Check your answer by substituting it back into the original equation. If both sides are equal, then you have found the solution. Remember to use order of operations

Multiple Choice

The goal of solving a one, two, or multi-step equations is to:

• Find the constant

• Find the coefficient

• Find the value of the variable

• Guess and check

Multiple Choice

True or False:

An equation can have more than one correct answer

• True

• False

Multiple Choice

What is the first step in solving this equation:
5x + 7 = 12

• Divide by 5

• Subtract 5

• Multiply by 7

• Subtract 7

Multiple Choice

What is the next step to solve this equation?
3x - 5 = 7

• Add 5 to both sides of the equation

• Add 3x to both sides of the equation

• Subtract 5 from both sides of the equation

• Subtract 7 from both sides of the equation

Multiple Choice

What is the next step to solve this equation?
-4x + 6 = 10

• Add 6 to both sides of the equation

• Add 10 to both sides of the equation

• Add 4x to both sides of the equation

• Subtract 6 from both sides of the equation

Multiple Choice

-54 = -4n + 2

• n = 13

• n = 14

• n = -56

• n = -14

Multiple Choice

-15 = -4m + 5

• 4

• 7

• 5

• 2.5

Multiple Choice

Solve:
3(x - 2) = 18

• 6.7

• 4

• 8

• -4

Multiple Choice

Solve the equation shown

• 51

• 52

• 27

• 21

Multiple Choice

7(2x - 4) = 42

• x= 5

• x= 8

• x= 2

• x= 9

Poll

Lesson Reflection

1 = I do not understand at all 5 = I understand completely

On a scale of 1 to 5, how well do you understand solving multi-step equations?

• 1

• 2

• 3

• 4

• 5

Inequalities

What is an inequality?

Review:

Equations (means-equal)

Remember when we solved equations we isolated the variable on one side by using inverse operations. You will solve inequalities the same way you solve equation. The only difference in them will be the symbol used.

How are equations and inequalities different?

Equations

Have an = sign

Only have one answer when solved

Inequalities

Don't have and = sign instead use inequality symbols

< (less than)

> (greater than)

≤ (less than or equal to)

≥ (greater than or equal to)

Can have multiple answers when solved

Example of an inequality and how to interpret your answer.

Multiple Choice

Solve the following inequality:

x + 2 > 12

• subtract 2 from both sides and your answer is

  x = 10

• subtract 2 from both sides and your answer is

  x > 10

• subtract 2 from both sides and your answer is

  X < 10

Multiple Choice

Solve 5x > 10

• divide by 5 on both sides and your answer is

  x = 2

• divide by 5 on both sides and your answer is

  x < 2

• divide by 5 on both sides and your answer is

  x > 2

Multiple Choice

Solve 3x + 2 > 8

• subtract 2 from both sides and your answer is

  x > 6

• subtract 2 from both sides and your answer is

  x = 6

• subtract 2 from both sides and then divide by 3 on both sides and your answer is

  x > 2

Multiple Choice

Solve the inequality

3x - 7 < 5

• x < 4

• x < -2

• x = 4

One Rule: If you are multiplying or dividing by a negative number you must switch the direction of the inequality symbol. 

• (ex) Solve -4x > -12

• Divide both sides by -4, and because we are dividing by a negative number, we must reverse the inequality sign

  ⇒ -4x/-4 < -12/-4

  ⇒ x < 3

Multiple Choice

Try some:

-2x ≤ 4

• divide by -2 o both sided and your answer will be

  x = -2

• divide by -2 on both sides and you will switch the direction of the inequality since you are dividing by a negative and your answer will be

  x ≥ -2

• divide by -2 on both sides and your answer will be

  x ≤ -2

Multiple Choice

- 3x < -12

• divide both sides by -3 and your answer is

  x < 4

• x = 4

• divide both sides by -3 and your answer is

  x > 4

  because you divided by a negative and had to flip the inequality symbol

Multiple Choice

Example

x ≥ 3

−2

• multiply both sides by -2 and your answer will be

  x ≤ -6

  because you multiplied by a negative you must switch the inequality

• multiply both sides by -2 and your answer will be

  x ≥ -6

• divide both sides by -2 and you get

  x ≥ -2/3