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How Do You Solve an AND Absolute Value Inequality and Graph It On a Number Line?

Solve this inequality for t and graph the solution set

Summary

  1. '<' means LESS THAN
  2. An absolute value with "LESS THAN" indicates an AND compound inequality
  3. '>' means GREATER THAN
  4. The inequality symbol flips when we multiply or divide by a negative number

Notes

    1. The '<' symbol means LESS THAN
    1. Two vertical bars indicate an absolute value
    1. This will give us our first scenario from the original absolute value compound inequality
    2. Two vertical bars are the absolute value symbols
    3. 74-0.8t < 10
    1. Remember, we determined this was an AND compound inequality!
    1. This will give us our second scenario from the original absolute value compound inequality
    2. 74-0.8t > -10
    1. 't' is just a variable we're trying to solve for
    1. Remember that a compound inequality means there are two inequalities, so we need to solve them individually first
    1. 10-74 = -64
    2. -0.8t < -64
    3. The '<' symbol means LESS THAN
    1. -64/-0.8 = 80
    1. t > 80
    2. With inequalities, if we divide or multiply by a negative number, the inequality sign needs to be flipped
    3. The '>' symbol means GREATER THAN
    1. Remember, we determined this was an AND compound inequality!
    1. -10-74 = -84
    2. -0.8t > -84
    3. The '>' symbol means GREATER THAN
    1. -84/-0.8 = 105
    1. t < 105
    2. With inequalities, if we divide or multiply by a negative number, the inequality sign needs to be flipped
    3. The '<' symbol means LESS THAN
    1. Our answers were 't>80' and 't<105'
    2. The '>' symbol means GREATER THAN
    3. The '<' symbol means LESS THAN
    1. So we start with '{t|'
    2. Opening up your set just means to start with a curly brace, '{'
    3. "Such that" is represented by '|'
    1. Our answers were 't>80' and 't<105'
    2. The '>' symbol means GREATER THAN
    3. The '<' symbol means LESS THAN
    1. Our answers were 't>80' and 't<105'
    1. Our inequalities are 't>80' and 't<105'
    1. Our inequalities are 't>80' and 't<105'
    1. Our inequalities are 't>80' and 't<105'
    2. Notice that the GREATER THAN symbol, '>', points to the right, the same direction as the arrow in our graph
    3. If we had a GREATER THAN OR EQUAL TO, or '≥', then we would have a closed circle
    1. Our inequalities are 't>80' and 't<105'
    2. Notice that the LESS THAN symbol, '<', points to the left, the same direction as the arrow in our graph
    3. If we had a LESS THAN OR EQUAL TO, or '≤', then we would have a closed circle
    1. Our inequalities are 't>80' and 't<105'
    1. The overlapping area is called the "intersection" of the two inequalities
    1. So the intersection of our two inequalities is all numbers between 80 and 105
    1. So the intersection of our two inequalities is all numbers between 80 and 105
    2. Note that 80 and 105 are NOT included in this intersection
    1. So the intersection of our two inequalities is all numbers between 80 and 105
    2. Note that 80 and 105 are NOT included in this intersection