Math Journal 10-16Rearrange the equation so that y is a function of x
1. 2.
Solve for x.
3. 4.
Math Journal 10-15Solve the formula for the given variable.
1. 2.
Solve for x.
3. 4.
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?
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
Example 1: Solve the inequalities.
7x + 19 > -2x + 55 6x + 22 < -3x + 31 + 2x+ 2x
9x+ 19> 55
- 19 -19
9x> 36
9 9
x > 4
+ 3x+ 3x
9x+ 22< 31
- 22 -22
9x< 9
9 9
x< 1
Example 2: Solve the inequalities.
x + 2 > 3x + 1 -8x + 7 < 4x – 5
- 3x- 3x
-2x + 2> 1
- 2- 2
-2x> -1
-2 -2
- 4x- 4x
-12x + 7< - 5
- 7- 7
-12x< -12
-12
x <2
1 x > 1
-12
Example 3: Solve the inequality.
(-12x + 16) < 10 – 3(-x – 2)
+ 4-3x < 10
-3x+ 4< 16
- 3x - 3x
-6x + 4< 16
+ 3x
- 4 - 4
-6x< 12
+ 6
+ 3x
-6
x > -2
4
1
-6
Example 4: Solve the inequality.
(12x – 4) < 2(7 – 5x)
- 26x < 14
+ 10x
16x - 2< 14
- 10x
+ 2 + 2
+ 10x
16
x < 1
16x< 16
2
1
16
Example 5: Solve the inequalities.
12 – 2a < - 5a – 9 x – 2x + 3 > 3 – x + 5a+ 5a
12+ 3a< - 9
- 12 - 12
3a< - 21
3 3
a < -7
- x+ 3> 3- x
+ x+ x
3> 3
true statementinfinite solutions
- 5x
< 5x- 25
- 5x
+ 245x
24< -25
false statementno solutions
6y- 3y
3y
+ 6> 5y- 4
- 5y - 5y
> 5y+ 6 - 4
- 6- 6- 4-2y + 6>
-2
Example 6: Solve the inequalities.
5x + 24 < 5(x - 5) 6y - (3y - 6) > 5y - 4
> -10 -2y
y < 5
-2
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?
.36 + .03x > .06x- .03x- .03x
.36 > .03x
12 > x
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.
.03 .03
x < 12
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
150 + 2x
+ 1x+ 1x
150 + 3x < 195- 150- 150
3x < 45
x < 15
Justin will weigh less than Tyson up until the 15
week mark.
< 195 - 1x
3 3
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.