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How do you solve and write the following in interval notation: #(2x + 3)(x − 1) ≥ 0#?

Answer 1

Avery Alexander

#x !in (-3/2, 1)#

Both (2x+3) and (x-1) should be either #>=0# or #<=0#

For #2x+3 >=0 and x-1>=0, x>=-3/2 and x>=1#..

Here, #x>=1# includes #x>=-3/2#.

Likewise, for the other case, #x<=-3/2#.

Combining both, #x !in (-3/2, 1)#..

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Answer 2

Eva Adamson

To solve and write the inequality (2x + 3)(x - 1) ≥ 0 in interval notation, you would follow these steps:

Find the critical points by setting each factor equal to zero and solving for x: 2x + 3 = 0 x - 1 = 0
Solve for x: 2x = -3 x = -3/2

x = 1
Plot these critical points on a number line, dividing the number line into intervals: -∞ --- (-3/2) --- 1 --- ∞
Test a value in each interval in the original inequality (2x + 3)(x - 1) ≥ 0 to determine the sign of the expression in each interval.
Write the solution in interval notation based on the signs: (-∞, -3/2] ∪ [1, ∞)

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.