1. Math Journal 10-16 −3𝑥+𝑦=7 2. 10𝑥 −5𝑦=25
Rearrange the equation so that y is a function of x
−3𝑥+𝑦= 𝑥 −5𝑦=25
Solve for x.
3. − 𝑥−1 +5𝑥=5𝑥 −2𝑥−5≥11

2. Math Journal 10-15 𝑚= 1 3 𝑛𝑡 𝑓𝑜𝑟 𝑛 2. 9 5 𝐶+32=𝐹 𝑓𝑜𝑟 𝐶
Solve the formula for the given variable.
𝑚= 1 3 𝑛𝑡 𝑓𝑜𝑟 𝑛 𝐶+32=𝐹 𝑓𝑜𝑟 𝐶
Solve for x.
3. −2 𝑥−3 +4𝑥=5𝑥− 𝑥−8≥13

3. Unit 3 Day 7: Solving Inequalities with Variables on Both Sides
Essential Questions: How do we solve inequalities with variables on both sides? When does an inequality have no solution or a solution of all real numbers?

4. Vocabulary
No solution: when the variable in an equation or inequality is eliminated and you are left with a false statement
All real numbers: when the variable in an equation or inequality is eliminated and you are left with a true statement

5. Example 1: Solve the inequalities.
7x + 19 > -2x x + 22 < -3x + 31
+ 2x
+ 2x
+ 3x
+ 3x 9x
+ 19
> 55 9x
+ 22
< 31
- 19
-19
- 22
-22 9x
> 36 9x
< 9 9 9 9 9 x
> 4 x
< 1

6. Example 2: Solve the inequalities.
x + 2 > 3x x + 7 < 4x – 5
- 3x
- 3x
- 4x
- 4x
-2x + 2
> 1
-12x + 7
< - 5
- 2
- 2
- 7
- 7
-2x
> -1
-12x
< -12 -2 -2
-12
-12
x < 2 1
x > 1

7. Example 3: Solve the inequality. (-12x + 16) < 10 – 3(-x – 2)4 1
-3x
+ 4
< 10
+ 3x
+ 6
-3x
+ 4
< 16
+ 3x
- 3x
- 3x
-6x + 4
< 16
- 4
- 4
-6x
< 12 -6 -6
x > -2

8. Example 4: Solve the inequality. (12x – 4) < 2(7 – 5x)
- 2
< 14
- 10x
+ 10x
+ 10x
16x
- 2
< 14
+ 2
+ 2
16x
< 16 16 16
x < 1

9. Example 5: Solve the inequalities.
12 – 2a < - 5a – 9 x – 2x + 3 > 3 – x
+ 5a
+ 5a
- x
+ 3
> 3
- x 12
+ 3a
< - 9
+ x
+ x
- 12
- 12 3
> 3 3a
< - 21
true statement
infinite solutions 3 3
a < -7

10. Example 6: Solve the inequalities.
5x + 24 < 5(x - 5) 6y - (3y - 6) > 5y - 4
+ 24 5x
< 5x
- 25 6y
- 3y
+ 6
> 5y
- 4
- 5x
- 5x 3y
> 5y
+ 6
- 4 24
< -25
- 5y
- 5y
- 4
-2y + 6
>
false statement
no solutions
- 6
- 6
> -10
-2y -2 -2
y < 5

11. .36 + .03x > .06x - .03x - .03x .36 > .03x .03 .03 12 > x
Example 7: Phone Company A charges an activation fee of 36 cents and then 3 cents per minute. Phone Company B charges 6 cents per minute with no activation fee. For what value of x is Phone Company A more expensive than Phone Company B?
x > .06x
Phone Company A is more expensive when the number of minutes is less than 12. If you talk for more than 12 minutes, Phone Company A is a good choice.
- .03x
- .03x
.36 > .03x
.03
.03
12 > x
x < 12

12. Justin will weigh less than Tyson up until the 15 week mark.
Example 8: Justin and Tyson are beginning an exercise program to train for football season. Justin weighs 150 pounds and hopes to gain 2 pounds per week. Tyson weighs 195 pounds and hopes to lose 1 pound per week. If the plan works, for how many weeks will Justin weigh less than Tyson?
Justin
Tyson x
< x
+ 1x
+ 1x
Justin will weigh less than Tyson up until the 15 week mark.
x < 195
- 150
- 150
3x < 45 3 3
x < 15

13. Summary
Essential Questions: How do we solve inequalities with variables on both sides? When does an inequality have no solution or a solution of all real numbers?
Take 1 minute to write 2 sentences answering the essential question.