Video: Solving Multistep Linear Inequalities with One Unknown on Both Sides

Solve the inequality 9𝑥 − 3(−7𝑥 + 9) < −7(−9 + 𝑥) − 2 in ℚ.

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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.

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