The solution set of the inequation `(-3)/2<1+x` is ______

Options

A. `{x//xlt(-5)/2}`
B. `{x//xgt(-5)/2}`
C. `{x//xge(-5)/2}`
D. `{x//xle(-5)/2}`

Answer

To solve the inequation \(-\frac{3}{2} < 1 + x\), we will follow these steps:

### Step 1: Isolate the variable \(x\)
We start with the given inequation:
\[ -\frac{3}{2} < 1 + x \]
To isolate \(x\), we need to move the constant term (which is \(1\)) to the left side. We do this by subtracting \(1\) from both sides of the inequation:
\[ -\frac{3}{2} - 1 < x \]

### Step 2: Simplify the left side
Next, we need to simplify the left side of the inequation. To do this, we convert \(1\) into a fraction with the same denominator as \(-\frac{3}{2}\):
\[ 1 = \frac{2}{2} \]
Now we can perform the subtraction:
\[ -\frac{3}{2} - \frac{2}{2} < x \]
This gives us:
\[ -\frac{3 + 2}{2} < x \]
\[ -\frac{5}{2} < x \]

### Step 3: Write the solution set
The inequation \(-\frac{5}{2} < x\) can be rewritten as:
\[ x > -\frac{5}{2} \]
Thus, the solution set is:
\[ \{ x \,|\, x > -\frac{5}{2} \} \]

### Final Answer
The solution set of the inequation \(-\frac{3}{2} < 1 + x\) is:
\[ \{ x \,|\, x > -\frac{5}{2} \} \]

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