Find all the value of x that satisfy at least one of the two inequalities 2x-7\le 1 or 2x+1>3

Question

Find all the value of x that satisfy at least one of the two inequalities 2x-7≤ 1 or 2x+1>3

Answer 1

Because the problem states that values need to satisfy at least one of the inequalities, we can simply take the union of solutions to each of the inequalities. This means we find all values of x that satisfy the first one and all values of x that satisfy the second one and then the combination of all those solutions is our final answer.

For the first:

2x - 7 ≤ 1

2x ≤ 8

x ≤ 4

So our solutions for the first are any values less than or equal to 4: (-∞, 4].

For the second:

2x + 1 > 3

2x > 2

x > 1

So our solutions for the second are any values greater than 1: (1, ∞).

The union, or combination, of these sets of values yields all real numbers (-∞, ∞). We get everything less than or equal to 4 from the first set and everything greater than 4 from the second set.

As an extra note, we actually get everything greater than 1 from our second set but that naturally overlaps a little bit with some of the solutions from the first set. For example, 3 is already counted as part of the first set since it's less than 4 but is also part of the second set since it's greater than 1. This overlap from 1 to 4 - in interval notation: (1, 4] - is called the intersection of the two sets. This would be the set of values that are solutions to both inequalities.