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How Do You Solve a Word Problem Using an AND Absolute Value Inequality?

Wanita played outside today and asked her mother if she could play outside tomorrow. Her mother said she could play outside only if the difference between today’s temperature and 80% of tomorrow's temperature is less than 10 degrees. If today’s temperature was 74 degrees what is the range of tomorrow’s temperature needed for Wanita to be able to play outside. Write the answer in set builder notation and graph the solution set on a number line.

Summary

  1. The | | symbols stand for absolute value
  2. '<' means 'less than' and '>' means 'greater than'
  3. Rewrite the inequality without the absolute value symbol
  4. Then rewrite that inequality, but make 10 negative, and flip the inequality symbol

Notes

    1. 't' is a variable representing the range of tomorrow's possible temperatures
    1. 't' is a variable representing the range of tomorrow's possible temperatures
    2. Range means a series of numbers
    1. If we were just looking for a number, an equality would be better
    1. 'The difference' means that we are looking for a number, and don't care about the sign of the number
    1. 'The difference' means that we are looking for just the value of the number
    1. 'The difference' means that we are looking for just the value of the number
    2. An absolute value finds the value, or magnitude, of a number
    3. The | | symbols stand for absolute value
    1. 'Difference' means that we are looking for an absolute value
    2. So today's temperature and 80% of tomorrows temperature are what goes inside the absolute value
    1. The | | symbols stand for absolute value
    2. '74 minus 0.8t' goes inside the absolute value symbols
    3. We are subtracting because we are finding a difference
    4. To convert a 80% to a decimal, we drop the % and divide by 100
    5. 't' is a variable representing the range of tomorrow's possible temperatures
    1. The '<' symbol means 'less than'
    2. The absolute value of what we have, 74 minus 0.8t, must be less than 10
    1. 't' is a variable representing the range of tomorrow's possible temperatures
    1. '74-0.8t' must be greater than -10 AND less than 10
    1. That is just 74 - 0.8t < 10
    1. It should have 74 - 0.8t on the left
    1. It should have 74 - 0.8t on the left, and then > -10
    1. Subtract 74 from '74 minus 0.8t is less than 10'
    1. 74 - 0.8t is on the left side, and positive 10 is on the right
    2. Subtract 74 from both sides
    3. We're left with -0.8t on the left, and -64 on the right
    1. Dividing the left side by -0.8 gets 't' by itself, and dividing the right side by -0.8 gets a positive 80
    1. We divided by -0.8, so the inequality becomes a greater than symbol, >
    1. Subtract 74 from '74 minus 0.8t is greater than -10'
    1. 74 - 0.8t is on the left side, and negative 10 is on the right
    2. Subtract 74 from both sides
    3. We're left with -0.8t on the left, and -84 on the right
    1. Dividing the left side by -0.8 gets 't' by itself, and dividing the right side by -0.8 gets a positive 105
    1. We divided by -0.8, so the inequality becomes a less than symbol, <
    1. 't' must be greater than 80 but also less than 105
    2. 't' is a variable representing the range of tomorrow's possible temperatures
    1. 't' is a variable representing the range of tomorrow's possible temperatures
    1. Our answer is 't>80 and t<105'
    2. '<' means 'less than' and '>' means 'greater than'
    3. 't' is a variable representing the range of tomorrow's possible temperatures
    1. '<' means 'less than' and '>' means 'greater than'
    2. 't' is a variable representing the range of tomorrow's possible temperatures
    1. In set-builder notation, our answer is {t|t>80 and t<105}
    1. Since 80 and 105 are in our compound inequality, it will be helpful if they are represented on our number line
    1. An open circle represents the fact that 't' cannot be 80 in the inequality 't>80'
    2. '>' means greater than
    3. 't' is a variable representing the range of tomorrow's possible temperatures
    1. Greater than means that the arrow should be pointing to the right
    2. 't' is a variable representing the range of tomorrow's possible temperatures
    1. An open circle represents the fact that 't' cannot be 105 in the inequality 't>105'
    2. 't' is a variable representing the range of tomorrow's possible temperatures
    1. Less than means that the arrow should be pointing to the left
    2. 't' is a variable representing the range of tomorrow's possible temperatures
    1. There are two arrows on the line graph, one pointing left and one point right
    1. There are two arrows on the line graph, one pointing left and one point right
    2. The two inequalities intersect where there are two colored lines visible on the number line
    1. After all, 't' must be both less than 105 and greater than 80
    1. After all, 't' must be both less than 105 and greater than 80
    2. We put open circles at 80 and 105 because they are not included in the intersection