Solve each inequality. Give the solution set in interval notation...

Question

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. (2x-5)/-8≤1-x

Accepted Answer

Hey, everyone. Welcome back in this problem. We are asked to find the solution set for the following inequality And were asked to express the answer and interval notation and the inequality were given is three X -1 divided by - Is less than or equal to 2 -6 x. Now let's just rewrite where we're given three X minus one over negative four is less than or equal to two minus six X. Now, when we have an inequality like this, and we're asked to find the solution set. What that really means is find the X values that satisfy this inequality. And in order to do that, we want to isolate X okay. So we're going to do that just like we do with an equation with an equal sign. We're going to get all of the X terms on one side and all of the constant terms on the other. So looking at our inequality, what we want to do first is deal with this negative four. Let's multiply both sides by -4, The left hand side becomes three X -1. Okay. On the right hand side, we get to minus six X times negative four. Now when we multiply by a negative and we're dealing with inequalities, we have to flip the inequality. So because we've multiplied by negative four, we now flip the inequality. So we have three, X minus one is greater than, or equal to two minus six X all times negative four, all right, three X minus one. If we expand on the right hand side, B at negative eight plus 24 X, again, we're trying to isolate for X. So let's move the three X to the right hand side and we'll move the negative eight to the left hand side. So on the left hand side, we have negative one plus eight, that's gonna give us seven. And on the right hand side, we have 24 X minus three X, that's going to give us 21 X. We can divide both sides by 21. Get that 7/21 is greater than, or equal to X. Okay. And we just want to write this the other way. So it's easier to read. We have X is less than or equal to. Now, seven and 21 both divide by seven. We can write this as one third. And so this is the solutions that we were looking for. Okay, any X value that is less than or equal to one third will satisfy this equation. Now, we're asked to write this in interval notation. And so we want X values less than or equal to 1/3. Okay. So we want the largest value to be 1/3 and we use a square bracket to indicate that we're including one third because of this equality. And this can go all the way down to negative infinity. There are no restrictions on the lower end. Okay. Now we use a round bracket with negative infinity and we never include the values at infinity. So always around bracket there. And that is the interval notation of the solutions that we found. And so if we look at our answer choices, we see that this corresponds with the answer choice. A, we have the interval from negative infinity up to and including one third. Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. (2x-5)/-8≤1-x

Key Concepts

Inequalities

Interval Notation

Solving Inequalities

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