For a constant k, the inequality \frac{k - 3x}{2} > \frac{5x + 3}{4} has exactly 4 positiv… — SAT

For a constant $k$ , the inequality $\frac{k - 3x}{2} > \frac{5x + 3}{4}$ has exactly 4 positive integer solutions for $x$ . If $k$ is an integer, how many possible values of $k$ are there?

Answer: 6

Answer

The number of possible integer values for

k

is 6.

Simplifying the inequality gives

x < \frac{2k - 3}{11}

. For this inequality to have exactly 4 positive integer solutions, the solution set must contain only the integers 1, 2, 3, and 4. This requires the boundary

\frac{2k - 3}{11}

to satisfy

4 < \frac{2k - 3}{11} \le 5

. Solving for

k

gives

23.5 < k \le 29

. The integers in this interval are 24, 25, 26, 27, 28, and 29, which total 6 values.

Step-by-Step Solution

Multiply both sides of the inequality by 4.

2(k - 3x) > 5x + 3

Clearing the denominators makes the inequality easier to solve.

Distribute the 2 on the left side.

2k - 6x > 5x + 3

Expanding terms allows us to group variables.

Add

6x

to both sides and subtract 3 from both sides.

11x < 2k - 3

Grouping

x

on one side and parameter terms on the other side.

Divide by 11.

x < \frac{2k - 3}{11}

Isolating

x

gives the boundary for the solution set.

Establish the range for the boundary term

\frac{2k - 3}{11}

4 < \frac{2k - 3}{11} \le 5

For the inequality to contain exactly the positive integers 1, 2, 3, and 4, the boundary must lie strictly above 4 and less than or equal to 5.

Solve the compound inequality for

k

23.5 < k \le 29

Performing algebraic operations on all parts of the inequality to isolate

k

Identify and count all integer solutions for

k

6 integers (24, 25, 26, 27, 28, 29)

Counting the integers in the range

(23.5, 29]

yields the final answer.

Key Concept

Solving linear inequalities in one variable with parameter constraints and identifying integer solution sets.

Estimated Time:3m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic