Question

QUESTION 1

The following is a student solution to the inequality

5/12-(x-3)/6≤(x-2)/3

5/12-2/2*(x-3)/6≤4/4*(x-2)/3

5/12-(2x-6)/12≤(4x-8)/12

5-2x-6≤4x-8

-1-2x≤4x-8

-6x≤-9

x≥3/2

There are two mathematical errors in this work. Identify at what step each mathematical error occurred and explain why it is mathematically incorrect.

The first mathematical error occurred going from line ____ to line ____.

Why it is incorrect:


The second mathematical error occurred going from line ____ to line ____.


QUESTION 2

Which of the following equations have the same solution? Give reasons for your answer that do not depend on solving the equations.
-(7-4x)=9

12=-4(-6x-3)

5x+34=-2(1-7x)

14=-(x-8)

-8=-(x+4)

x+5=-5x+5

Why it is incorrect:



b. Solve the inequality correctly.

Answer 1

QUESTION 1

The first mathematical error occurred going from line 3 to line 4.

Why it is incorrect: distributive property of multiplication is not well applied

-(2x-6) = -2x+6

The second mathematical error occurred going from line 5 to line 6.

Why it is incorrect: the "-1" should had passed as "+1" from the left side to the right side of the inequality

-1-2x≤4x-8

-2x-4x≤-8+1

-6x≤-7

QUESTION 2

To answer this question we have to rewrite the equations in a similar way, as follows:

-(7-4x)=9

7-4x = -9

-4x+16=0 (eq. 1)

12=-4(-6x-3)

12=24x+12

0=24x (eq. 2)

5x+34=-2(1-7x)

5x+34=-2+14x

-9x+36 = 0 (eq. 3)

if you multiply equation 1 by 9/4, you get equation 3, then they have the same solution

14=-(x-8)

14=-x+8

x+6=0 (eq. 4)

-8=-(x+4)

-8=-x-4)

x-4 = 0 (eq. 5)

if you divide equation 1 by -4, you get equation 5, then they have the same solution

x+5=-5x+5

6x=0 (eq. 6)

if you divide equation 2 by 4, you get equation 6, then they have the same solution.

b.

5/12-(x-3)/6≤(x-2)/3

5/12-2/2*(x-3)/6≤4/4*(x-2)/3

5/12-(2x-6)/12≤(4x-8)/12

5-2x+6≤4x-8

11-2x≤4x-8

-6x≤-19

x≥19/6

Answer 2

1)

1st Error: In going from Step 3 to Step 3.

Reason: Negative sign is not distributed inside the brackets.

2nd Error: In going from Step 5 to Step 6

Reason: Sign of the number is not changed while moving to other side of inequality,

2)

a) 12=-4(-6x-3) and x+5=-5x+5

b) -(7-4x)=9 and 5x+34=-2(1-7x) and -8=-(x+4)

Explanation:

Question 1)

The given inequality is:

`(5)/(12)-(x-3)/(6) \leq  (x-2)/(3)`

Step 1: Making the denominators common for all fractions

`(5)/(12)-(2)/(2) * (x-3)/(6) \leq (4)/(4) * (x-2)/(3)`

This step is done correctly in the given solution.

Step 2: Simplifying

`(5)/(12)-(2x-6)/(12)\leq (4x-8)/(12)`

This step is done correctly in the given solution

Step 3: Multiplying both sides by 12, and simplifying.

`5-(2x-6)\leq 4x-8\\\\ 5-2x+6\leq 4x-8`

First error is made in this step. While opening the brackets, the negative sign should be distributed inside the bracket, which will change the signs.

Step 4: Simplification:

`11-2x\leq 4x-8`

Step 5: Moving Common terms to one side and simplifying

`-2x-4x\leq -8-11\\\\ -6x\leq -19`

Error was made in this step. When a number is moved to other side, its sign will be changed.

Step 6: Dividing both sides by -6

`x\geq (19)/(6)`

Conclusion:

1st Error: In going from Step 3 to Step 3.

Reason: Negative sign is not distributed inside the brackets.

2nd Error: In going from Step 5 to Step 6

Reason: Sign of the number is not changed while moving to other side of inequality,

Question 2:

In the Equation 2: 12=-4(-6x-3), when -4 will be multiplied inside the brackets, the 12 on eft hand side will cancel the 12 that will appear on right hand side, giving a result that will lead to x = 0.

Same is the case with Equation 6: x+5=-5x+5, 5 on both sides will cancel out leaving x = 0.

So, 2nd and 6th equations will have the same solution.

In Equation 1, on expanding the bracket and moving 7 to other side, we get a relation: 4x = 16

In Equation 3, on simplifying the right hand side, and carrying common terms to one side, we get the relation: - 9x = -36

In Equation 5, on expanding the bracket and simplifying the relation is reduced to 4 = x

It can be observed that all these 3 equations have the same solution i.e. x = 4

So, the following set of Equations have the same solution:

a) 12=-4(-6x-3) and x+5=-5x+5

b) -(7-4x)=9 and 5x+34=-2(1-7x) and -8=-(x+4)