Video Transcript
Solve the inequality six 𝑥 minus
27 over four is greater than or equal to four-fifths in the set of rational
numbers.
To solve this inequality in the set
of rational numbers, we need to use the axioms of inequalities to isolate the
variable on one side of the inequality. We can start by multiplying both
sides of the inequality by the two denominators of four and five to eliminate the
fractions. When multiplying both sides of an
inequality by a constant, we must remember to reverse the direction of the
inequality if the value we multiply by is negative. However, this doesn’t apply
here. So we obtain five multiplied by six
𝑥 minus 27 is greater than or equal to four times four.
Distributing the five over the
parentheses on the left-hand side and evaluating the constant on the right-hand side
gives 30𝑥 minus 135 is greater than or equal to 16. Next, we isolate the 𝑥-term on the
left-hand side by adding 135 to both sides of the inequality. This gives the equivalent
inequality 30𝑥 is greater than or equal to 151.
Finally, we divide both sides of
the inequality by 30 to obtain 𝑥 is greater than or equal to 151 over 30, again a
reminder that when we divide both sides of an inequality by a negative value, we
must reverse the direction of the inequality. But this doesn’t apply here as the
value we divided by was positive.
We were asked to solve this
inequality in the set of rational numbers. So we can express our solution in
set notation as the set of all values of 𝑥 such that 𝑥 is a rational number and 𝑥
is greater than or equal to 151 over 30.
