Question Video: Identifying an Equivalent Linear Inequality Mathematics • First Year of Preparatory School

Video Transcript

Which of the following inequalities is equivalent to the inequality five 𝑛 minus nine multiplied by 𝑛 plus three is less than 14? (A) Four 𝑛 is less than 11. (B) Four 𝑛 is less than 41. (C) Negative four 𝑛 is less than 11. (D) Negative four 𝑛 is less than 17. (E) Negative four 𝑛 is less than 41.

We’ve been given an inequality in the variable 𝑛 and asked to determine which of five simpler inequalities it is equivalent to. To answer this question, let’s work through the process of solving this inequality. We won’t get all the way to a solution, but at some point, we should arrive at an inequality that matches one of the five options.

The first step in simplifying this inequality is to distribute the negative nine over the parentheses on the left-hand side. Doing so gives five 𝑛 minus nine 𝑛 minus 27 is less than 14. We then have a choice of what to do next, but probably the most sensible step is to combine the terms in 𝑛. Five 𝑛 minus nine 𝑛 is negative four 𝑛. This inequality doesn’t yet match any of the five options, so we need to keep going.

The next step is to isolate the term in 𝑛 by adding 27 to each side of the inequality. We know this is valid because adding or indeed subtracting a constant to or from both sides of an inequality gives an equivalent inequality. Simplifying the constant on the right-hand side gives 41, so the entire inequality becomes negative four 𝑛 is less than 41. At this point, we see that this is option (E), so we’ve found the inequality which is equivalent to the starting one.

It’s worth pointing out that there are actually other equivalent inequalities we could arrive at if we continued solving this inequality. If we were to divide both sides of the inequality by negative one, we would obtain four 𝑛 on the left-hand side and negative 41 on the right. But importantly, we must remember that if we multiply or divide both sides of an inequality by a negative value, we have to reverse the direction of the inequality sign. So the equivalent inequality would be four 𝑛 is greater than negative 41.

Of course, we could also divide both sides of the inequality by four to actually solve it and find that 𝑛 is greater than negative 41 over four. This time, we don’t reverse the direction of the inequality as the value we’re dividing by is positive. Our answer is option (E). The equivalent inequality is negative four 𝑛 is less than 41.