(4x-1)(x^2-4)<0

To solve the inequality \((4x-1)(x^2-4)<0\), we need to identify the points where the expression equals 0 and analyze the intervals between those points to determine where the expression is negative.

Step 1: Find the zeros of the expression

1. Set each factor equal to zero:

   • \(4x - 1 = 0\) \[ 4x = 1 \implies x = \frac{1}{4} \]

   • \(x^2 - 4 = 0\) can be factored as: \[ x^2 - 4 = (x - 2)(x + 2) = 0 \] This gives us the solutions \(x = 2\) and \(x = -2\).

Step 2: Identify critical points

The critical points derived from the factors are:

• \(x = -2\)
• \(x = \frac{1}{4}\)
• \(x = 2\)

Step 3: Test intervals determined by the critical points

Now we will divide the number line into intervals based on the critical points and test each interval:

1. Intervals:

   • \((- \infty, -2)\)
   • \((-2, \frac{1}{4})\)
   • \((\frac{1}{4}, 2)\)
   • \((2, \infty)\)

2. Choose test points:

   • For \((- \infty, -2)\), choose \(x = -3\): \[ (4(-3)-1)((-3)^2-4) = (-12-1)(9-4) = (-13)(5) = -65 < 0 \]

   • For \((-2, \frac{1}{4})\), choose \(x = 0\): \[ (4(0)-1)(0^2-4) = (-1)(-4) = 4 > 0 \]

   • For \((\frac{1}{4}, 2)\), choose \(x = 1\): \[ (4(1)-1)(1^2-4) = (4-1)(1-4) = (3)(-3) = -9 < 0 \]

   • For \((2, \infty)\), choose \(x = 3\): \[ (4(3)-1)(3^2-4) = (12-1)(9-4) = (11)(5) = 55 > 0 \]

Step 4: Combine results

The expression is negative in the following intervals:

• \((- \infty, -2)\)
• \((\frac{1}{4}, 2)\)

Step 5: Write the final solution

The solution to the inequality \((4x-1)(x^2-4)<0\) is: \[ (-\infty, -2) \cup \left(\frac{1}{4}, 2\right) \]