Problem 49 - Beginning and Intermediate Algebra

Question

Solve each inequality, and graph the solution set. $$ \frac{4 k}{2 k-1}

Step-by-Step Solution

Answer

The solution is \( k \) in the interval \( (0, \frac{5}{2}) \cup ( \frac{5}{2}, \infty ) \).

Step 1: Clear the fraction

Multiply both sides of the inequality by \(2k - 1\) to eliminate the fraction: \[ \frac{4k}{2k-1} (2k - 1) < k(2k - 1) \] This simplifies to \[ 4k < k(2k - 1) \]

Step 2: Distribute on the right

Distribute the \(k\) on the right side: \[ 4k < 2k^2 - k \]

Step 3: Bring all terms to one side

Subtract \(4k\) from both sides to set the inequality to 0: \[ 0 < 2k^2 - 5k \]

Step 4: Factor the quadratic expression

Factor out the greatest common factor: \[ 0 < k(2k - 5) \]

Step 5: Analyze the factors

Analyze the factors \(k(2k - 5)\) to determine where the product is positive. Set each factor to 0: \[ k = 0 \] \[ 2k - 5 = 0 \] Solving these gives \[ k = 0 \] and \[ k = \frac{5}{2} \].

Step 6: Test intervals

Using the critical points \(k = 0\) and \(k = \frac{5}{2}\), test the intervals \((-\infty, 0)\), \( (0, \frac{5}{2} )\), and \((\frac{5}{2}, \infty)\) to see where the inequality holds true.

Step 7: Determine solution

The product \(k(2k - 5) > 0 \) will be positive in the intervals \( (0, \frac{5}{2} )\) and \( (\frac{5}{2}, \infty) \). Therefore, the solution to the inequality is \( k > 0 \) and \( k eq \frac{1}{2} \).

Step 8: Graph the solution set

Graph the solution set on a number line. There will be an open circle at \( 0 \) and at \( \frac{5}{2} \), shading the interval between them as well as the region to the right of \( \frac{5}{2} \).

Key Concepts

Graphing Inequalities

Solving inequalities often involves finding a range of values that satisfy the given condition. To visualize these solutions, we can graph them on a number line.
First, solve the inequality by isolating the variable and determining critical points, which are the values that make the inequality switch from true to false.
After finding these points, test intervals around them to determine where the inequality holds true.
Use open circles for points not included in the solution (strict inequalities like < or >) and closed circles for points included (≤ or ≥).
Finally, shade the regions that satisfy the inequality.

Factoring Quadratic Expressions

Factoring is crucial when solving quadratic inequalities. It involves rewriting the quadratic expression as a product of its factors.
For instance, in the inequality (0 < k(2k - 5)):

• Identify the common factor: k
• Rewrite as: (k(2k - 5))

Factoring makes it easier to identify the critical points, which are solutions where the expression equals zero.
This turns the quadratic problem into a simpler problem of finding where the product of factors is positive or negative.

Inequality Intervals

Inequality intervals help identify the range of values that solve an inequality.
After finding the critical points, divide the number line into intervals based on these points.
For our inequality, these intervals are: (-∞, 0), (0, 5/2), and (5/2, ∞).
Test a value from each interval in the original inequality:

• For (-∞, 0), choose k=-1
• For (0, 5/2), choose k=1
• For (5/2, ∞), choose k=3

Check if these values satisfy the inequality to determine where the solution lies.

Critical Points

Critical points are values where the expression equals zero. They separate different intervals for testing.
For example, k=0 and k=5/2 are critical points from solving k(2k-5)=0.
Critical points help to determine where the expression changes from positive to negative or vice versa.
Use these points to test intervals and find the solution set.
These points are essential for sketching the graph of the solution on a number line.

Quadratic Inequality

Quadratic inequalities involve expressions of the form ax^2 + bx + c < 0 or > 0.
To solve quadratic inequalities:

• Get the inequality in standard quadratic form: 0 < 2k^2 - 5k
• Factor the quadratic expression: 0 < k(2k-5)
• Determine the critical points: k=0 and k=5/2
• Analyze intervals around critical points
• Identify where the inequality holds true

Graph the solution to visualize the interval where the inequality is satisfied.
In this case, the solution intervals are (0, 5/2) and (5/2, ∞).
Ensure your graph is accurate and clearly shows the solution set.