Question Find the value of $c$ that gives the equation $|x - 4| + 3 = c$ exactly one solution.

Studdy Solution

STEP 1

Assumptions
1. The equation given is $c = |x - 4| + 3$ .
2. We are looking for the value of $c$ that would result in the equation having exactly one solution.

STEP 2

Understand the properties of absolute value functions. An absolute value function $|x - a|$ has two cases:
1. $x - a$ when $x \geq a$
2. $-(x - a)$ when $x < a$

STEP 3

The equation $|x - 4| + 3 = c$ will have exactly one solution when the expression inside the absolute value is equal to zero, because this is the point at which the function changes direction.

STEP 4

Set the expression inside the absolute value equal to zero to find the point where the function changes direction.
$x - 4 = 0$

STEP 5

Solve for $x$ .
$x = 4$

STEP 6

Now that we know the function changes direction at $x = 4$ , we can find the value of $c$ when $x = 4$ .

STEP 7

Substitute $x = 4$ into the original equation to find $c$ .
$c = |4 - 4| + 3$

STEP 8

Calculate the absolute value.
$c = |0| + 3$

STEP 9

Since the absolute value of zero is zero, we simplify the equation.
$c = 0 + 3$

STEP 10

Calculate the value of $c$ .
$c = 3$
The value of $C$ that would cause the equation to have exactly 1 solution is $3$ .
\textbf{A.} \quad 3

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Question Find the value of $c$ that gives the equation $|x - 4| + 3 = c$ exactly one solution.

Studdy Solution

STEP 1

Assumptions
1. The equation given is $c = |x - 4| + 3$ .
2. We are looking for the value of $c$ that would result in the equation having exactly one solution.

STEP 2

Understand the properties of absolute value functions. An absolute value function $|x - a|$ has two cases:
1. $x - a$ when $x \geq a$
2. $-(x - a)$ when $x < a$

STEP 3

The equation $|x - 4| + 3 = c$ will have exactly one solution when the expression inside the absolute value is equal to zero, because this is the point at which the function changes direction.

STEP 4

Set the expression inside the absolute value equal to zero to find the point where the function changes direction.
$x - 4 = 0$

STEP 5

Solve for $x$ .
$x = 4$

STEP 6

Now that we know the function changes direction at $x = 4$ , we can find the value of $c$ when $x = 4$ .

STEP 7

Substitute $x = 4$ into the original equation to find $c$ .
$c = |4 - 4| + 3$

STEP 8

Calculate the absolute value.
$c = |0| + 3$

STEP 9

Since the absolute value of zero is zero, we simplify the equation.
$c = 0 + 3$

STEP 10

Calculate the value of $c$ .
$c = 3$
The value of $C$ that would cause the equation to have exactly 1 solution is $3$ .
\textbf{A.} \quad 3

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Question Find the value of ccc that gives the equation ∣x−4∣+3=c|x - 4| + 3 = c∣x−4∣+3=c exactly one solution.

Studdy Solution

STEP 1

Assumptions1. The equation given is c=∣x−4∣+3c = |x - 4| + 3c=∣x−4∣+3.2. We are looking for the value of ccc that would result in the equation having exactly one solution.

STEP 2

Understand the properties of absolute value functions. An absolute value function ∣x−a∣|x - a|∣x−a∣ has two cases:1. x−ax - ax−a when x≥ax \geq ax≥a2. −(x−a)-(x - a)−(x−a) when x<ax < ax<a

STEP 3

The equation ∣x−4∣+3=c|x - 4| + 3 = c∣x−4∣+3=c will have exactly one solution when the expression inside the absolute value is equal to zero, because this is the point at which the function changes direction.

STEP 4

Set the expression inside the absolute value equal to zero to find the point where the function changes direction.x−4=0x - 4 = 0x−4=0

STEP 5

Solve for xxx.x=4x = 4x=4

STEP 6

Now that we know the function changes direction at x=4x = 4x=4, we can find the value of ccc when x=4x = 4x=4.

STEP 7

Substitute x=4x = 4x=4 into the original equation to find ccc.c=∣4−4∣+3c = |4 - 4| + 3c=∣4−4∣+3

STEP 8

Calculate the absolute value.c=∣0∣+3c = |0| + 3c=∣0∣+3

STEP 9

Since the absolute value of zero is zero, we simplify the equation.c=0+3c = 0 + 3c=0+3

STEP 10

Calculate the value of ccc.c=3c = 3c=3The value of CCC that would cause the equation to have exactly 1 solution is 333.\textbf{A.} \quad 3

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Question Find the value of $c$ that gives the equation $|x - 4| + 3 = c$ exactly one solution.

Assumptions
1. The equation given is $c = |x - 4| + 3$ .
2. We are looking for the value of $c$ that would result in the equation having exactly one solution.

Understand the properties of absolute value functions. An absolute value function $|x - a|$ has two cases:
1. $x - a$ when $x \geq a$
2. $-(x - a)$ when $x < a$

The equation $|x - 4| + 3 = c$ will have exactly one solution when the expression inside the absolute value is equal to zero, because this is the point at which the function changes direction.

Set the expression inside the absolute value equal to zero to find the point where the function changes direction.
$x - 4 = 0$

Solve for $x$ .
$x = 4$

Now that we know the function changes direction at $x = 4$ , we can find the value of $c$ when $x = 4$ .

Substitute $x = 4$ into the original equation to find $c$ .
$c = |4 - 4| + 3$

Calculate the absolute value.
$c = |0| + 3$

Since the absolute value of zero is zero, we simplify the equation.
$c = 0 + 3$

Calculate the value of $c$ .
$c = 3$
The value of $C$ that would cause the equation to have exactly 1 solution is $3$ .
\textbf{A.} \quad 3