How do you solve #(2x-1)(x+2)(x-4)

Question

How do you solve #(2x-1)(x+2)(x-4)<=0#?

Isaiah Anthony · Accepted Answer

The inequality holds for #x\le -2# and #1/2\le x \le 4#.

Keep in mind the rule for multiplying: Combining these rules, you can clearly see that a product of factors is negative if and only if there is an odd number of negative factors (in this case, one or all three). Now, #2x-1# is positive if and only if #x>1/2#, #x+2# is positive if and only if #x> -2#, and #x-4# is positive if and only if #x>4#. So, the important points are #-2#, #1/2# and #4#. Before #-2#, all three factors are negative, so the product is negative. Between #-2# and #1/2#, #x+2# is positive and the other two are negative, so the product is positive. Between #1/2# and #4#, only #x-4# is negative, and the other two are positive, so the product is negative. After #4#, all the factors are positive, so the product is positive.

Eva Adamson · Answer

undefined To solve the inequality (2x - 1)(x + 2)(x - 4) ≤ 0, you need to find the intervals on the number line where the expression is less than or equal to zero. This involves identifying the critical points where the expression equals zero and determining the sign of the expression in each interval between these critical points. The critical points occur where each factor equals zero, so:

1. Set each factor equal to zero:
   - 2x - 1 = 0 ⇒ x = 1/2
   - x + 2 = 0 ⇒ x = -2
   - x - 4 = 0 ⇒ x = 4

2. Plot these critical points on a number line and divide the line into intervals:
   - Interval 1: (-∞, -2)
   - Interval 2: (-2, 1/2)
   - Interval 3: (1/2, 4)
   - Interval 4: (4, ∞)

3. Test a value from each interval into the expression (2x - 1)(x + 2)(x - 4):
   - For Interval 1, choose x = -3: (-) * (-) * (-) = -
   - For Interval 2, choose x = 0: (+) * (+) * (-) = -
   - For Interval 3, choose x = 3: (+) * (+) * (+) = +
   - For Interval 4, choose x = 5: (+) * (+) * (+) = +

4. Analyze the signs:
   - In Interval 1 and Interval 2, the expression is negative or zero, satisfying the inequality.
   - In Interval 3 and Interval 4, the expression is positive, not satisfying the inequality.

5. Combine the intervals where the expression is negative or zero:
   - Interval (-∞, -2] ∪ [-2, 1/2]

Therefore, the solution to the inequality (2x - 1)(x + 2)(x - 4) ≤ 0 is: 
x ∈ (-∞, -2] ∪ [-2, 1/2].

How do you solve #(2x-1)(x+2)(x-4)&lt;=0#?

How do you solve #(2x-1)(x+2)(x-4)<=0#?