Video Transcript
Does 𝑥 equals 13 over two satisfy the inequality six 𝑥 minus 27 over four is greater than or equal to four-fifths?
We’ve been asked whether a particular value of 𝑥 satisfies a given inequality. One way to answer this question would be to substitute the given value of 𝑥 into the inequality and determine whether it gives a true statement. But this looks like it’s going to involve multiplying and dividing some reasonably complicated fractions. So, instead, let’s solve the inequality and then determine whether this particular value of 𝑥 is in the solution set.
We’ll begin by multiplying both sides of the inequality by the two denominators of four and five to eliminate the fractions. This gives five multiplied by six 𝑥 minus 27 is greater than or equal to four multiplied by four.
Remember that when we multiply or divide both sides of an inequality by a negative value, this reverses the direction of the inequality. But this doesn’t apply here as the values of four and five are both positive. Next, we distribute the five over the parentheses on the left-hand side and simplify the constant on the right-hand side to give the equivalent inequality 30𝑥 minus 135 is greater than or equal to 16. We can then isolate the 𝑥-term by adding 135 to both sides of the inequality to give 30𝑥 is greater than or equal to 151. Finally, we divide both sides of the inequality by 30 to give 𝑥 is greater than or equal to 151 over 30. We don’t need to be concerned about reversing the inequality here as the value we’re dividing by is positive.
So we’ve solved the inequality. And now we need to determine whether the value of 13 over two is in the solution set. The question becomes, is 13 over two greater than or equal to 151 over 30? To answer this question, let’s convert the two fractions to have a common denominator. Two is a factor of 30. So multiplying both the numerator and denominator by 15 gives that 13 over two is equivalent to 195 over 30. As the denominators of the two fractions are now the same, we just have to compare the numerators. And so clearly 195 over 30 is greater than 151 over 30. So, 13 over two is certainly greater than or equal to 151 over 30, which means that the value 𝑥 equals 13 over two does satisfy the inequality six 𝑥 minus 27 over four is greater than or equal to four-fifths. So, our answer is yes.
