Question

QUESTION 1

Answer 1

Answer:

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)

Step-by-step explanation:

Question 1)

The given inequality is:

$\frac{5}{12}-\frac{x-3}{6} \leq \frac{x-2}{3}$

Step 1: Making the denominators common for all fractions

$\frac{5}{12}-\frac{2}{2} \times \frac{x-3}{6} \leq \frac{4}{4} \times \frac{x-2}{3}$

This step is done correctly in the given solution.

Step 2: Simplifying

$\frac{5}{12}-\frac{2x-6}{12}\leq \frac{4x-8}{12}$

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

$x\geq \frac{19}{6}$

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)

Answer 2

Answer:

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