Question

Please Help Me.

1)Sam solved the following problem:


-4x + 5 > 29

-4x + 5 - 5 > 29 - 5

-4x > -24

-4x/-4 > -24/-4

x > 6


What were his two mistakes?


2)Hillary buys 3 pounds of Gala apples and some Granny Smith apples. Both kinds of apples cost $4.50 per pound. The total cost is $24.75. How many pounds of Granny Smith apples did Hillary buy? Write and solve a variable equation. Show your work.


3)Tell whether each equation has one solution, infinitely many solutions, or no solution.


a)6x + 8 = 6(x + 2)

b)10x = 15 + 5x

c)x + 11 = 8x + 11 - 7x

Answer 1

1) First mistake: 29 - 5 = 24, not -24

Second mistake: Division by negative number changes the sign of inequality. Sam didn't change the sign.

2) 2.5 kg

3) a) no solutions

b) one solution

c) infinitely many solutions

Explanation:

1)Sam solved the following problem:

`-4x+5>29\\ \\-4x+5-5>29-5\ [\text{Sam used the addition property}]\\ \\-4x>24\ [\text{Here Sam made a mistake, because }29-5=24\text{ not }-24]\\ \\(-4x)/(-4)<(24)/(-4)\ [\text{Here Sam made the second mistake}]`

Dividing by negative number changes the sign of the inequality.

Thus,

`x<-6`

2) Hillary buys 3 pounds of Gala apples which cost $4.50 per pound and paid

`\$4.50\cdot 3=\$13.50`

Let x be the number of pounds of Granny Smith apples Hillary bought. She paid $4.50x for x pounds of Granny Smith apples.

The total cost of Hillary buying is $24.75, so

`\$4.50x+\$13.50=\$24.75`

Solve this equation

`\$4.50x=\$24.75-\$13.50\\ \\4.50x=11.25\\ \\x=2.5\ kg`

of Granny Smith apples.

3) Consider all equations:

a)
`6x+8=6(x+2)`

Use distributive property:

`6x+8=6x+12\\ \\6x+8-6x=6x+12-6x\\ \\8=12`

You get false equality, so this equation has no solutions.

b)
`10x=15+5x`

Add -5x:

`10x-5x=15+5x-5x\\ \\5x=15\\ \\x=3\ [\text{Divide by 5}]`

This equation has one solution

c)
`x+11=8x+11-7x`

This equation is equivalent to

`x+11=(8x-7x)+11\\ \\x+11=x+11\\ \\x+11-11=x+11-11\\ \\x=x\\ \\x-x=x-x\\ \\0=0`

You get the equality that is true for all values of x, so the equation has infinitely many solutions.