If `5 (4 -x) - 4 lt 3x - 2 gt 4x -6,` then the value of x is (a) 3 (b) 1 (c) 2 (d) 5

Options

A. 3
B. 1
C. 2
D. 5

Answer

To solve the inequality \(5(4 - x) - 4 < 3x - 2 > 4x - 6\), we can break it down into two separate inequalities:

1. \(5(4 - x) - 4 < 3x - 2\)
2. \(3x - 2 > 4x - 6\)

### Step 1: Solve the first inequality

Starting with the first inequality:

\[ 5(4 - x) - 4 < 3x - 2 \]

Distributing \(5\):

\[ 20 - 5x - 4 < 3x - 2 \]

Simplifying the left side:

\[ 16 - 5x < 3x - 2 \]

Now, we will add \(5x\) to both sides:

\[ 16 < 3x + 5x - 2 \]

Combining like terms:

\[ 16 < 8x - 2 \]

Next, we will add \(2\) to both sides:

\[ 16 + 2 < 8x \]

This simplifies to:

\[ 18 < 8x \]

Dividing both sides by \(8\):

\[ \frac{18}{8} < x \]

Which simplifies to:

\[ \frac{9}{4} < x \quad \text{or} \quad x > 2.25 \]

### Step 2: Solve the second inequality

Now, we will solve the second inequality:

\[ 3x - 2 > 4x - 6 \]

Subtract \(3x\) from both sides:

\[ -2 > 4x - 3x - 6 \]

This simplifies to:

\[ -2 > x - 6 \]

Adding \(6\) to both sides:

\[ 4 > x \quad \text{or} \quad x < 4 \]

### Step 3: Combine the results

From the two inequalities, we have:

1. \(x > \frac{9}{4}\) (which is approximately \(2.25\))
2. \(x < 4\)

Combining these gives us:

\[ \frac{9}{4} < x < 4 \]

### Step 4: Determine the possible values of \(x\)

Now we check the options provided:

(a) \(3\)
(b) \(1\)
(c) \(2\)
(d) \(5\)

The only value that satisfies \(\frac{9}{4} < x < 4\) is:

**(a) 3**

### Final Answer:

The value of \(x\) is \(3\).

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