### Video Transcript

Solve the inequality nine 𝑥 minus
three multiplied by negative seven 𝑥 plus nine is less than negative seven
multiplied by negative nine plus 𝑥 minus two in the set of rational numbers.

So now as you might remember from
set notation, this ℚ means rational numbers. Rational numbers are numbers that
can be represented by a fraction.

Now to solve this problem, what we
need to do, first of all, is distribute across our parentheses on both sides of our
inequality. And that’s because we’re gonna
solve it in the same way we’d solve an equation. So on the left-hand side, we’re
gonna get nine 𝑥 plus, and then we’ve got 21𝑥. This is positive 21𝑥 because we’ve
got negative three multiplied by negative seven 𝑥. And then we have minus 27. And this gonna be less than 63. And that’s because negative seven
multiplied by negative nine gives us 63 cause a negative multiplied by a negative
gives us a positive. And then we’ve got minus seven 𝑥
minus two.

Okay great, so now what we need to
do is simplify both sides. So when we do that, we’re gonna get
30𝑥 minus 27 is less than 61 minus seven 𝑥. So now the next step is to get the
𝑥s all on one side of the inequality and numerical values on the other. So what I’m gonna do to do that is
add seven 𝑥 to each side of the inequality and add 27 to each side of the
inequality. So when we do this, we’re gonna get
37𝑥 is less than 88.

So finally, all that’s left to do
is divide each side of the inequality by 37. When we do that, we get 𝑥 is less
than 88 over 37. And then we can show that using our
interval notation such a way that 𝑥 is a member of the set of rational numbers such
that 𝑥 is less than 88 over 37.