Question

The solution set of the system of inequalities \( 5-4x \leq -7 \) or \( 5-4x \geq 7,\ x \in R \) is

• A. \( (-\infty,-\frac{1}{2}) \cap [3,\infty) \)
• B. \( (-\infty,-\frac{1}{2}) \cup (3,\infty) \)
• C. \( (-\infty,-\frac{1}{2}] \cap (3,\infty) \)
• D. \( (-\infty,-\frac{1}{2}] \cup [3,\infty) \)

Hint:

When solving inequalities, remember that dividing or multiplying by a negative number reverses the inequality sign. For compound inequalities joined by “or”, take the union of the solution sets.

Answer 1

The Correct Option is D

Step 1: Write the given compound inequality.
The given system of inequalities is:
\[ 5-4x \leq -7 \quad \text{or} \quad 5-4x \geq 7. \]
Since the word used is \(\text{or}\), we will find the solution of both inequalities separately and then take their union.

Step 2: Solve the first inequality.
Consider:
\[ 5-4x \leq -7. \]
Subtracting \(5\) from both sides, we get:
\[ -4x \leq -12. \]

Step 3: Divide by the negative coefficient carefully.
Now divide both sides by \(-4\). Since we divide by a negative number, the inequality sign will reverse:
\[ x \geq 3. \]
So, the solution of the first inequality is:
\[ [3,\infty). \]

Step 4: Solve the second inequality.
Now consider:
\[ 5-4x \geq 7. \]
Subtracting \(5\) from both sides, we get:
\[ -4x \geq 2. \]

Step 5: Divide by the negative coefficient again.
Divide both sides by \(-4\). Again, the inequality sign will reverse:
\[ x \leq -\frac{1}{2}. \]
So, the solution of the second inequality is:
\[ (-\infty,-\frac{1}{2}]. \]

Step 6: Combine the two solution sets.
Since the connector is \(\text{or}\), we take the union of both solution sets:
\[ x \leq -\frac{1}{2} \quad \text{or} \quad x \geq 3. \]
Therefore, the complete solution set is:
\[ (-\infty,-\frac{1}{2}] \cup [3,\infty). \]

Step 7: Match with the given options.
The solution set:
\[ (-\infty,-\frac{1}{2}] \cup [3,\infty) \]
matches option (D).
Final Answer:
\[ \boxed{(-\infty,-\frac{1}{2}] \cup [3,\infty)}. \]