Respuesta :
The number line for [tex]\fbox{\begin\\\ \bf \text{option 1}\\\end{minispace}}[/tex] represents the solution for the equation [tex]|x+4|=2[/tex].
Further explanation:
The equation is given as follows:
[tex]|x+4|=2[/tex]
In the above equation [tex]||[/tex] represents the modulus function.
Modulus function is defined as a function which gives positive value of the function for any real value of [tex]x[/tex].
For example: The function [tex]y=|x|[/tex] is a modulus function in which [tex]y>0[/tex] and [tex]x<0[/tex] or [tex]x>0[/tex].
In the given equation [tex]|x+4|[/tex] is a modulus expression.
There are two cases formed for [tex]|x+4|[/tex].
First case: [tex]x>-4[/tex]
If [tex]x>-4[/tex] then [tex]|x+4|\rightarrow (x+4)[/tex].
Substitute [tex]|x+4|=(x+4)[/tex] in [tex]|x+4|=2[/tex].
[tex]\begin{aligned}x+4&=2\\x&=2-4\\&=-2\end{aligned}[/tex]
Therefore, for [tex]x>-4[/tex] the value of [tex]x[/tex] which satisfies the equation [tex]|x+4|=2[/tex] is [tex]x=-2[/tex].
Second case: [tex]x<-4[/tex]
If [tex]x<-4[/tex] then [tex]|x+4|\rightarrow -(x+4)[/tex].
Substitute [tex]|x+4|=-(x+4)[/tex] in [tex]|x+4|=2[/tex].
[tex]\begin{aligned}-(x+4)&=2\\-x-4&=2\\-x&=2+4\\-x&=6\\x&=-6\end{aligned}[/tex]
Therefore, for [tex]x<-4[/tex] the value of [tex]x[/tex] which satisfies the equation [tex]|x+4|=2[/tex] is [tex]x=-6[/tex].
This implies that the solution for the equation [tex]|x+4|=2[/tex] or the value of [tex]x[/tex] which satisfies the given equation are [tex]\fbox{\begin\\\ \math x=-2\ \text{and}\ x=-6\\\end{minispace}}[/tex].
Option 1:
The number line in option 1 shows that the solutions of the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-6[/tex].
As per our calculation made above the solutions for the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-6[/tex].
This implies that option 1 is correct.
Option 2:
The number line in option 2 shows that the solutions of the equation [tex]|x+4|=2[/tex] are [tex]x=2[/tex] and [tex]x=4[/tex].
As per our calculation made above the solutions for the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-6[/tex].
This implies that option 2 is incorrect.
Option 3:
The number line in option 3 shows that the solutions of the equation [tex]|x+4|=2[/tex] are [tex]x=2[/tex] and [tex]x=6[/tex].
As per our calculation made above the solutions for the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-6[/tex].
This implies that option 3 is incorrect.
Option 4:
The number line in option 4 shows that the solutions of the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-4[/tex].
As per our calculation made above the solutions for the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-6[/tex].
This implies that option 4 is incorrect.
Therefore, the number line for [tex]\fbox{\begin\\\ \bf \text{option 1}\\\end{minispace}}[/tex] represents the solution for the equation [tex]|x+4|=2[/tex].
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Answer details:
Grade: High school
Subject: Mathematics
Chapter: Functions
Keywords: Functions, modulus, modulus function, number line, real line, |x+4|=2, equation, root, zeroes, solutions, absolute function, x=-6 and x=-2.
Option A is correct, the values of [tex]x[/tex] staisfying the given equation [tex]|x+4|=2[/tex] are [tex]-2[/tex] and [tex]-6[/tex].
Modulus function always returns a positive value to the equation.
Here, [tex]|x+4|[/tex] will give positive result if [tex]x>-4[/tex] and negative value if, [tex]x<-4[/tex].
Case 1: When [tex]x>-4[/tex]
According to the given equation,
[tex]|x+4|=2\\x=2-4\\x=-2[/tex]
So, the value of [tex]x[/tex] which satisfies the given equation [tex]|x+4|=2[/tex] for [tex]x>-4[/tex] is [tex]x=-2[/tex].
Case 2: When [tex]x<-4[/tex]
According to the given equation,
[tex]|x+4|=2\\-x-4=2\\x=-6[/tex]
So, the value of [tex]x[/tex] which satisfies the given equation [tex]|x+4|=2[/tex] for [tex]x<-4[/tex] is [tex]x=-6[/tex].
Hence, the values of [tex]x[/tex] staisfying the given equation [tex]|x+4|=2[/tex] are [tex]-2[/tex] and [tex]-6[/tex].
Now, according to the options, Option A is correct.
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