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What Does It Mean When an Inequality is a Contradiction or Has No Solution?

What does it mean when an inequality has no solution or is a contradiction?

Summary

  1. '<' means 'less than'
  2. 4 is NEVER less than 2 so 4 < 2 is a false statement
  3. Subtract 'x' from both sides of x+4 < x+2 and you're left with 4 < 2, which is false
  4. 'x' is a variable here
  5. '>' means 'greater than'
  6. Subtract '6x' from both sides of 10+6x > 15+6x and you're left with 10 > 15, which is also false
  7. No matter what we plug in for 'x' in the second two examples, we get false statements
  8. The solution set is the empty set and the number line is empty when we have no solution

Notes

    1. Before learning the definition of an inequality with no solution, let's look at some examples
    1. First, look at an example that only involves numbers
    2. '<' means 'less than'
    3. This inequality is obviously false because 4 is NOT less than 2
    1. Next, let's look at an inequality with variables on each side
    2. 'x' is the variable here
    3. '<' means 'less than'
    4. Subtract 'x' from both sides and you're left with the inequality we saw in the first example
    5. A 'contradiction' is another word for a false statement
    1. Finally, let's look at a slightly more complicated inequality with variables on each side
    2. 'x' is the variable here
    3. '>' means 'greater than'
    4. To test this inequality, try plugging 2 in for 'x'
    5. The little question mark above the inequality sign means we don't know yet whether this inequality will give us a true statement
    6. Use the order of operations to simplify
    7. 22 is NOT greater than 27, so plugging in 2 for 'x' gives us a false statement
    8. Try plugging 6 in for 'x'
    9. 46 is NOT greater than 51, so plugging in 6 for 'x' also gives us a false statement
    1. 'x' is the variable here
    2. '>' means 'greater than'
    3. Just subtract '6x' from both sides to get 10 > 15
    4. A 'contradiction' is another word for a false statement
    5. We know that 10 is definitely not greater than 15, so this gives us a false statement
    1. We've seen some examples, now let's go over the definition of an inequality with no solution
    2. We chose 2 and 6 as our values for 'x', but we could have chosen any numbers
    3. We still would have gotten a false statement!
    4. The 'empty set' is a set with no values in it
    5. Try graphing an inequality with no solution on a number line -- it will be empty!