What is the distance between x-intercepts of the graph of the equation

distance between x-intercepts of the graph of the equation y = 2|4x – 4| – 10
=> ± 2|4x – 4| – 10

When slope is positive: and Y = 0
y = 2(4x – 4) – 10
y = 8x – 8 – 10
0 = 8x - 18
18 = 8X
9/4 = X --------------(1)

When slope is negative: and Y = 0
y = -2(4x – 4) – 10
y = -8x + 8 – 10
0 = -8x - 2
2 = -8X
-1/4 = X --------------(2)

Distance between two values
=> 9/4 - (- 1/4)
=> 9/4+1/4
=> 10/4
=> 5/2

Answer is D

Posted from my mobile device

x - intercept means, y = 0;

Given the equation of a straight line, y = 2 |4x – 4| – 10

Implies y = 2 (4x - 4) -10 ------and----- y = - 2 (4x - 4) -10

Putting y = 0

=> 2 (4x - 4) -10 = 0 ---- and ---- 2 (4x - 4) + 10 = 0

=> 4x - 4 - 5 = 0 ---- and ---- 4x - 4 + 5 = 0

=> 4x = 9 ---- and ---- 4x = - 1

=> x = \(\frac{9}{4}\) ---- and ---- x = - \(\frac{1}{4}\)

Required distance between x - intercepts = \(\frac{9}{4}\) + \(\frac{1}{4}\) = \(\frac{5}{2}\)

IMO ans is D.

Hi,
At x intercept points of the line y will be 0.
So, we need to simplify equation by substituting y=0.

0=2|4x-4|-10
|4x-4|=5
Now, (4x-4)=5 ----1
and, -(4x-4)=5 ----2
solving both equations, we get x=9/4 from eq.1 and x= -1/4 from eq.2
now the distance between both the points will be x1-x2= (9/4)-(-1/4)
= 10/4 or 5/2 which is option D.

The X Intercepts will occur in the Coordinate Plane for the Graph of this function when Y = 0


Scenario 1: when the Quantity Inside the Modulus is greater than or equal to >/= 0

Rule: for X > 0, the Absolute Value of X becomes: | X | = X

0 = 2 * +(4X - 4) - 10

10 = 8X - 8

X = 18/8 = 9/4



Coordinates for 1st X Intercept are:

( 9/4 , 0)



Scenario 2: when the Quantity Inside the Modulus is Less Than < 0

Rule: when X < 0, then the Absolute Value of X becomes: | X | = -(X)

***The reason why is because the OUTPUT from an Absolute Value Modulus must ALWAYS be NON-Negative. Thus, we must Negate the (-)Negative Inner Quantity when we Open the Modulus to make it (+)Positive


Again, the X Intercept will cross the x-Axis when Y = 0


0 = 2 * -(4X - 4) - 10

10 = -8X + 8

X = -(2/8) = -(1/4)



Coordinates for the 2nd X-Intercept are:

( -1/4 , 0)


Lastly: The Distance between the X-Intercepts is the Absolute Distance between the X Coordinates of each X-Intercept


(9/4) - (-1/4) =

10/4 =

5/2

Answer -D-

Posted from my mobile device

Important property: The x-intercepts are the points where the y-coordinate = 0.
So we'll take the given equation: y = 2|4x – 4| – 10
And replace y with 0 to get: 0 = 2|4x – 4| – 10
Add 10 to both sides of the equation: 10 = 2|4x – 4|
Divide both sides of the equation by 2 to get: 5 = |4x – 4|

Another property: If |x| = k, then x = k or x = -k
Apply this property: either 4x – 4 = 5 or 4x – 4 = -5

If 4x – 4 = 5, then x = 9/4, which means the first x-intercept is at (9/4, 0)
If 4x – 4 = -5, then x = -1/4, which means the second x-intercept is at (-1/4, 0)

So, the distance between the two x-intercepts = 9/4 - (-1/4) = 10/4 = 5/2

Answer: D

Since the expression is an absolute value expression, two values of x can satisfy the expression when it is equated to zero.

Why do we equate the expression to zero? Because we are trying to find the x-intercepts of the equation.
Graphically, the x-intercept is the point on the x-axis where the line/curve representing an equation intersects the x-axis. Therefore, at this point, the value of y is equal to zero.

Equating the expression to zero, we have 2|4x – 4| – 10 = 0

2 |4x – 4| = 10

|4x – 4| = 5

This means that 4x – 4 = 5 or 4x – 4 = -5.

Therefore, x = \(\frac{9}{4}\) or x = -\(\frac{1}{4}\)

So, the graph of the equation, y = 2|4x -4| -10 intersects the x-axis at (-\(\frac{1}{4}\), 0) and (\(\frac{9}{4}\),0).
The distance between the x-intercepts = \(\frac{9}{4}\) – (-\(\frac{1}{4}\)) = \(\frac{10 }{4}\) = \(\frac{5}{2}\).

The correct answer option is D.

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