Video: Solving Linear Inequalities with One Unknown on Both Sides

Solve the inequality 10𝑥 + 16 ≤ 8(𝑥 − 19) in ℚ.

02:18

Video Transcript

Solve the inequality 10𝑥 plus 16 is less than or equal to eight multiplied by 𝑥 minus 19 in the set of rational numbers.

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 inequality.

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.

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