Inequality with all real numbers as solutions

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.

Compound inequality with all real numbers as 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.

Compound inequality with all real numbers as solutions

Absolute value inequality with all real numbers as solutions

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.

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