Linear Inequalities In One Variable

We start by solving the inequality: To eliminate the fractions, we find the least common denominator (LCD), which is 12: Simplifying the equation: Multiplying both sides by 12: Dividing both sides by 7: Thus, the solution is .

Step 1: Understanding the Concept:
means .
Step 2: Detailed Explanation:

Subtract 3:
Divide by -4 (reverse inequality signs):
Thus .
Step 3: Final Answer:
.

Concept: Find the range of using given bounds of .

Step 1: Use interval transformation.
Multiply by 5: Add 4:

Step 2: Compare with given form.
Thus,

Concept: Solve compound inequalities step-by-step.

Step 1: Split inequality.


Step 2: Solve first inequality.


Step 3: Solve second inequality.


Step 4: Combine results.

Concept:
• Solve compound inequality involving modulus

Step 1: Break inequality


Step 2: Solve individually


Step 3: Take intersection
Final Conclusion:
Option (D)

Step 1: Understanding the problem.
The word "BANANA" consists of the letters B, A, N, A, N, A. To form words starting and ending with the same letter, the first and last positions must be occupied by the same letter.
Step 2: Case breakdown.
We will break the problem into cases based on the letter at the start and end: Case 1: First and last letter is 'A'.
The remaining letters are B, A, N, N. These can be arranged in ways, as there are two N's. Case 2: First and last letter is 'N'.
The remaining letters are B, A, A, A. These can be arranged in ways, as there are three A's.
Step 3: Total number of arrangements.
The total number of arrangements is .
Final Answer: 72.

Step 1: Find the general term in the binomial expansion of .
The binomial expansion of is given by: We are looking for the coefficient of , which corresponds to the term where . The coefficient of is: We are given that this coefficient is 160, so:
Step 2: Solve for .
We need to solve the equation for . Begin by testing values of until we find one that satisfies the equation. After trying different values of , we find that .
Step 3: Find the coefficient of in .
Next, we look at the expansion of . The general term in the expansion is: We want the coefficient of , so we set , which gives . The coefficient of is:
Step 4: Calculate the coefficient.
Now, calculate the coefficient:
Step 5: Conclusion.
The coefficient of in is: