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

Solve this compound inequality for t: 74 - 0.8t < 10 AND 6 - 5t ≥ -519
Write the answer in set-builder notation and graph the solution set on a number line.

Summary

  1. The 'AND' means it's an AND compound inequality
  2. The inequality symbol flips when we multiply or divide by a negative number
  3. Open circles represent '<' or '>' symbols
  4. Notice that an arrow points in the same direction as the inequality symbol!
  5. Closed circles represent '≤' or '≥' symbols

Notes

    1. 74-0.8t < 10 is our first inequality
    2. 't' is just a variable we want to solve for
    3. We know this is a compound inequality because there are two inequalities
    1. Subtracting '74' from the left side will cancel out the '74' and give us '-0.8t' by itself
    2. Subtracting '74' from '10' on the right side will give us '-64'
    3. 10-74 = -64
    4. -0.8t < -64
    1. Dividing the left side by '-0.8' will cancel out the multiplication by '-0.8', leaving us with 't' alone
    2. Dividing '-64' on the right side by '-0.8' will give us '80'
    3. -64/-0.8 = 80
    1. Our first inequality is now 't>80'
    2. With inequalities, if we divide or multiply by a negative number, the inequality sign needs to be flipped
    1. Our second inequality is '74-0.8t > -10'
    2. 't' is just a variable we want to solve for
    3. We know this is a compound inequality because there are two inequalities
    1. Subtracting '6' from the left side will cancel out the '6' and give us '-5t' by itself
    2. Subtracting '6' from '-519' on the right side will give us '-525'
    3. -519-6 = -525
    4. -5t ≥ -525
    1. The 'AND' means we have an AND compound inequality
    1. Dividing the left side by '-5' will cancel out the multiplication by '-5', leaving us with 't' alone
    2. Dividing '-525' on the right side by '-5' will give us '105'
    3. -525/-5 = 105
    1. Our second inequality is now 't≤105'
    2. With inequalities, if we divide or multiply by a negative number, the inequality sign needs to be flipped
    1. The 'AND' means we have an AND compound inequality
    1. Our answer was the compound inequality, 't>80' AND 't≤105'
    1. Opening up your set just means to start with a curly brace, '{'
    2. "Such that" is represented by '|'
    3. So we start with '{t|'
    1. Our answer was the compound inequality, 't>80' AND 't≤105'
    2. The '>' symbol means GREATER THAN
    3. The '≤' symbol means LESS THAN OR EQUAL TO
    4. Don't forget to close your set with a curly brace, '}'
    1. Our answer was the compound inequality, 't>80' AND 't≤105'
    1. Our answer was the compound inequality, 't>80' AND 't≤105'
    1. Our answer was the compound inequality, 't>80' AND 't≤105'
    1. Our answer was the compound inequality, 't>80' AND 't≤105'
    2. The '>' symbol means GREATER THAN
    3. Open circles represent '<' or '>' symbols
    4. Notice that the GREATER THAN symbol, '>', points to the right, the same direction as the arrow in our graph
    1. Our answer was the compound inequality, 't>80' AND 't≤105'
    2. The '≤' symbol means LESS THAN OR EQUAL TO
    3. Closed circles represent '≤' or '≥' symbols
    4. Notice that the LESS THAN OR EQUAL TO symbol, '≤', points to the left, the same direction as the arrow in our graph
    1. Our answer was the compound inequality, 't>80' AND 't≤105'
    1. The overlapping area is called the "intersection"
    1. The overlapping area is called the "intersection"
    1. The overlapping area is called the "intersection"
    1. Open circles represent '<' or '>' symbols
    2. Closed circles represent '≤' or '≥' symbols
    3. The thicker line, or shading, represents the intersection of the two inequalities
    1. Our answers were the inequalities 't>80' and 't≤105'