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:
[tex]\frac{5}{12}-\frac{x-3}{6} \leq \frac{x-2}{3}[/tex]
Step 1: Making the denominators common for all fractions
[tex]\frac{5}{12}-\frac{2}{2} \times \frac{x-3}{6} \leq \frac{4}{4} \times \frac{x-2}{3}[/tex]
This step is done correctly in the given solution.
Step 2: Simplifying
[tex]\frac{5}{12}-\frac{2x-6}{12}\leq \frac{4x-8}{12}[/tex]
This step is done correctly in the given solution
Step 3: Multiplying both sides by 12, and simplifying.
[tex]5-(2x-6)\leq 4x-8\\\\ 5-2x+6\leq 4x-8[/tex]
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:
[tex]11-2x\leq 4x-8[/tex]
Step 5: Moving Common terms to one side and simplifying
[tex]-2x-4x\leq -8-11\\\\ -6x\leq -19[/tex]
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
[tex]x\geq \frac{19}{6}[/tex]
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:
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
