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How Do You Solve an Inequality Where You're Multiplying Positive Fractions?

Solve the inequality for h:
(3/5)h ≤ 42

Summary

  1. Multiplying '3/5' by it's reciprocal 5/3 will give us 'h' by itself
  2. Since we multiplied the left side by 5/3, we need to multiply the right side by 5/3, which gives us '42•(5/3)', or '70'
  3. (5/3)•(3/5)h = 1h = h
  4. The inequality does NOT flip since we didn't multiply or divide by a negative number
  5. We can write our answer in set-builder notation, as well

Notes

    1. We're trying to get 'h' by itself on the left side
    2. 'h' is a variable that we want to solve for
    1. This property states that whatever we multiply one side of the inequality by, we must multiply the other side by
    1. Multiplying by a reciprocal is the same as performing the opposite operation
    1. So to find the reciprocal of '3/5', we just flip the numerator and denominator and get 5/3
    1. Multiplying by a reciprocal is the same as performing the opposite operation
    2. 5/3 is the reciprocal of '3/5'
    1. Whatever we multiply one side of the inequality by, we must multiply the other side by
    2. So we multiply both sides by 5/3
    1. The left-hand side of the inequality now looks like this: (5/3)•(3/5)h
    1. The left-hand side of the inequality now looks like this: (5/3)•(3/5)h
    2. (5/3)•(3/5)h = 1h = h
    1. The right-hand side of the inequality now looks like this: 42•(5/3)
    2. 42•(5/3) = (42/1)•(5/3) = (42•5)/(1•3)
    3. 42•5 = 210
    4. 1•3 = 3
    5. 210/3 = 70
    1. As long as we didn't multiply by a negative number, we don't have to flip the inequality symbol
    2. 5/3 is NOT a negative number
    1. As long as we didn't multiply by a negative number, we don't have to flip the inequality symbol
    2. 5/3 is NOT a negative number
    1. The inequality 'h≤70' is our answer
    1. The inequality 'h≤70' is our answer
    1. Opening up a set just means we need to draw a left curly brace, '{'
    2. Curly braces tell us we're dealing with a set
    3. Here, 'h' is representing the set of numbers defined by the inequality we got for our answer
    1. So we put our answer after '{h |'
    2. The inequality 'h≤70' is our answer
    3. This gives us '{h | h≤70'
    1. To close a set, just draw a right curly brace, '}'
    2. Our answer in set-builder notation is '{h | h≤70}'