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Solution - Linear inequalities with one unknown

\frac{1}{x} < \frac{1}{90}

1/x<1/90

See steps

Step-by-step explanation

1. Group all constants on the right side of the inequality

$\frac{1}{x} + \frac{1}{60} < \frac{1}{36}$

Subtract $\frac{1}{60}$ from both sides:

$(\frac{1}{x} + \frac{1}{60}) - \frac{1}{60} < (\frac{1}{36}) - \frac{1}{60}$

Group like terms:

$(\frac{1}{60} + \frac{- 1}{60}) + \frac{1}{x} < (\frac{1}{36}) - \frac{1}{60}$

Combine the fractions:

$\frac{(1 - 1)}{60} + \frac{1}{x} < (\frac{1}{36}) - \frac{1}{60}$

Combine the numerators:

$\frac{0}{60} + \frac{1}{x} < (\frac{1}{36}) - \frac{1}{60}$

Reduce the zero numerator:

$0 + \frac{1}{x} < (\frac{1}{36}) - \frac{1}{60}$

Simplify the arithmetic:

$\frac{1}{x} < (\frac{1}{36}) - \frac{1}{60}$

Find the lowest common denominator:

$\frac{1}{x} < \frac{(1 \cdot 5)}{(36 \cdot 5)} + \frac{(- 1 \cdot 3)}{(60 \cdot 3)}$

Multiply the denominators:

$\frac{1}{x} < \frac{(1 \cdot 5)}{180} + \frac{(- 1 \cdot 3)}{180}$

Multiply the numerators:

$\frac{1}{x} < \frac{5}{180} + \frac{- 3}{180}$

Combine the fractions:

$\frac{1}{x} < \frac{(5 - 3)}{180}$

Combine the numerators:

$\frac{1}{x} < \frac{2}{180}$

Find the greatest common factor of the numerator and denominator:

$\frac{1}{x} < \frac{(1 \cdot 2)}{(90 \cdot 2)}$

Factor out and cancel the greatest common factor:

$\frac{1}{x} < \frac{1}{90}$

2. Plot the solution on a coordinate grid

Solution:
$\frac{1}{x} < \frac{1}{90}$

Interval notation:
$(- \infty, 1 / 90)$

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Why learn this

Inequalities help us understand how systems work by setting boundaries. For example, a speed limit of 30 miles per hour does not mean we have to drive exactly 30 miles per hour and, instead, establishes a boundary for what is allowable — drive more than 30 miles per hour and risk getting a ticket. This could be modelled mathematically as $x \leq 30$ .
There are also situations where there is more than one boundary. In our speed limit example, there may also be a lower speed limit of 15 miles per hour to prevent drivers from driving too slowly. The two boundaries together could be modelled mathematically as $15 \leq x \leq 30$ , in which $x$ represents all of the possible values between or equal to 15 and/or 30.

Furthermore, anytime we say something along the lines of, "it will take at least twenty minutes to get there," or "the car can hold five people at most," we are expressing the numerical boundaries of something and, therefore, speaking in terms of inequalities.

Terms and topics

Linear inequalities