An inequality with all real numbers as solutions is easy to solve or easy to recognize. See a few examples below.
Example #1
Solve x - x > -1
x - x > -1
x - x = 0, so we get 0 > -1
Since 0 is always bigger than -1, this inequality is always true. Therefore, all real numbers are solutions.
Example #2
Solve 5x + x + 3 > 6x + -4
5x + x + 3 > 6x + -4
Subtract 3 from each side of the inequality
5x + x + 3 - 3 > 6x + -4 - 3
5x + x + 0 > 6x + -4 + -3
5x + x > 6x + -7
Simplify by adding 5x and x
6x > 6x + -7
Subtract 6x from each side of the inequality
6x - 6x > 6x - 6x + -7
0 > 0 + -7
0 > -7
Since 0 is always bigger than -7, this inequality is always true. Therefore, all real numbers are solutions.
Example #3
Solve 8x + 4 ≤ 20 or 3x - 2 > 1
Solve 8x + 4 ≤ 20
8x + 4 ≤ 20
Subtract 4 from each side of the inequality
8x + 4 - 4 ≤ 20 - 4
8x ≤ 16
Divide each side by 8
8x/8 ≤ 16/8
x ≤ 2
Solve 3x - 2 > 1
3x - 2 > 1
Add 2 to each side of the inequality
3x - 2 + 2 > 1 + 2
3x - 2 + 2 > 1 + 2
3x > 3
Divide each side by 3
3x/3 > 3/3
x > 1
The solution is x ≤ 2 or x > 1 and this includes all real numbers as shown in the graph below.
Example #4
Solve |x| > -10
The absolute value of a number is always positive.
For example, suppose x is positive and choose x = 2
|2| = 2 and 2 is always bigger -10
Suppose x is negative and choose x = -2
|-2| = 2 and 2 is always bigger than -10
Therefore, |x| > -10 has all real numbers as solutions
No matter what number you choose for x, when you take the absolute value, it will always be bigger than -10.
In general, if |x| > a and a is a negative number, |x| > a has all real numbers as solutions.
Mar 29, 23 10:19 AM
Mar 15, 23 07:45 AM