Solve the inequality 10𝑥 plus 16
is less than or equal to eight multiplied by 𝑥 minus 19 in the set of rational
And as we said when reading out the
question, this ℚ means rational numbers. And what we mean by rational
numbers are any numbers that can be represented using a fraction. Okay, so now let’s solve our
Well, the first step or stage in
our multistep inequality is to distribute across our parentheses. And what this means is we’re gonna
multiply the eight by both terms inside our parentheses. So we’re gonna get 10𝑥 plus 16 is
less than or equal to. Then we’ve got eight 𝑥 cause eight
multiplied by 𝑥 is eight 𝑥. And then we’re gonna get minus
152. And that’s cause eight multiplied
by negative 19 gives this negative 152.
So now what we can do to make sure
that we have the 𝑥s on one side of our inequality and numerical values on the other
side of our inequality is to subtract eight 𝑥 and 16 from each side. So when I do this, what I’m gonna
get is two 𝑥 is less than or equal to negative 168. And then all we need to do is
divide both sides of the inequality by two. And we get 𝑥 is less than or equal
to negative 84.
So what this means is, 𝑥 is any
value that’s less than negative 84 but also including negative 84. Well, we’ve just shown the answer
using inequality notation cause 𝑥 is less than or equal to negative 84. We could also use some interval
notation as I’ve shown here, where we’ve got a parenthesis. Then we’ve got negative ∞, then
comma negative 84. And then we’ve got a square
bracket. And what this means is, the values
can take any value from negative ∞, but not including negative ∞, all the way up to
negative 84, and including negative 84.
And then we also have this other
way of representing our answer that tells us that 𝑥 is an element or a member of
the rational numbers such that 𝑥 is less than or equal to negative 84.