The simultaneous equations, `y=x+2|x| & y=4+x-|x|` have the solution set given by: (a) `(4/3,4/3)` b) `(4,4/3)` (c) `(-4/3,4/3)` (d) `(4/3,4)`

Options

A. `(4/3,4/3)`
B. `(4,4/3)`
C. `(-4/3,4/3)`
D. `(4/3,4)`

Answer

To solve the simultaneous equations \( y = x + 2|x| \) and \( y = 4 + x - |x| \), we will consider two cases based on the definition of the absolute value function.

### Step 1: Identify Cases for Absolute Value
The absolute value function \( |x| \) can be defined as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

### Step 2: Case 1: \( x \geq 0 \)
In this case, we can rewrite the equations:
1. \( y = x + 2|x| = x + 2x = 3x \)
2. \( y = 4 + x - |x| = 4 + x - x = 4 \)

Now we set the two expressions for \( y \) equal to each other:
\[ 3x = 4 \]
Solving for \( x \):
\[ x = \frac{4}{3} \]

Substituting \( x \) back into either equation to find \( y \):
\[ y = 3x = 3 \times \frac{4}{3} = 4 \]

Thus, for \( x \geq 0 \), we have the solution:
\[ \left( \frac{4}{3}, 4 \right) \]

### Step 3: Case 2: \( x < 0 \)
In this case, we rewrite the equations:
1. \( y = x + 2|x| = x + 2(-x) = x - 2x = -x \)
2. \( y = 4 + x - |x| = 4 + x - (-x) = 4 + x + x = 4 + 2x \)

Now we set the two expressions for \( y \) equal to each other:
\[ -x = 4 + 2x \]
Rearranging gives:
\[ -x - 2x = 4 \implies -3x = 4 \implies x = -\frac{4}{3} \]

Substituting \( x \) back into either equation to find \( y \):
\[ y = -x = -\left(-\frac{4}{3}\right) = \frac{4}{3} \]

Thus, for \( x < 0 \), we have the solution:
\[ \left(-\frac{4}{3}, \frac{4}{3}\right) \]

### Step 4: Summary of Solutions
The solutions we found are:
1. \( \left( \frac{4}{3}, 4 \right) \)
2. \( \left(-\frac{4}{3}, \frac{4}{3}\right) \)

### Step 5: Identify Correct Options
From the given options:
- (a) \( \left( \frac{4}{3}, \frac{4}{3} \right) \)
- (b) \( (4, \frac{4}{3}) \)
- (c) \( \left(-\frac{4}{3}, \frac{4}{3}\right) \)
- (d) \( \left( \frac{4}{3}, 4 \right) \)

The correct solutions are:
- \( \left( \frac{4}{3}, 4 \right) \) matches option (d)
- \( \left(-\frac{4}{3}, \frac{4}{3}\right) \) matches option (c)

### Final Answer
The solution set includes:
- \( \left( \frac{4}{3}, 4 \right) \) (option d)
- \( \left(-\frac{4}{3}, \frac{4}{3}\right) \) (option c)