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. 6x-(2x+3)≥4x-5

Accepted Answer

Hey, everyone in this problem, we're asked to find the solution set for the following inequality. And to express the answer in interval notation, we have seven X minus the factor two X plus five. It's greater than, or equal to five, X minus 11. We have four answer choices. Option A is the interval from zero to infinity. Option B is the interval from negative infinity to zero. Option C is the interval from negative infinity to positive infinity. And option D is the union of the intervals from negative infinity to zero with the interval from zero to positive infinity. And all of those intervals and the answer choices do not include the end points. So let's get started by writing out what we were given. We have seven X minus this two X plus five binomial is greater than, or equal to five X minus 11. So just because we have an inequality, we don't want to think about this differently. We don't want to change what we're doing. We wanna rearrange this just like we would with an equation, we wanna get X on one side and all of the numbers on the other side. So the first thing we're gonna do is expand the bracket on the left hand side. So we have seven X and then we have minus everything in brackets. Ok? So that minus that negative needs to apply to both terms. We get seven X minus two, X minus five is greater than, or equal to five X minus 11. Now, we're gonna move all of the X terms to the left, all of the numbers to the right and so on the right, we have five X in order to move it to the left, we need to subtract. So we get seven X minus two, X minus five X on the left-hand side. And then before we had this minus five term on the left to move it to the right, we're gonna add it. So we have that this is greater than or equal to negative plus five. Now simplifying seven X minus two, X minus five X. Well, that just gives a zero. So we get zero is greater than or equal to. And on the right hand side, we have negative 11 plus five which is gonna be negative six. We see that we've come to this inequality with just numbers, OK? When you come to a point like this, don't be worried, think about what it's telling you and try to figure out if the statement is true or false. OK? In this case, we have that zero is greater than or equal to negative six. Well, that's true. And we know that zero is bigger than negative six. That's true. And that's always true. OK? No matter what X value we substitute into this inequality, we are going to get this true statement out of it. OK? So this is always true which tells us that any X value from negative infinity up to positive infinity is going to be a solution. OK? So the solution set here is gonna be the interval from negative infinity up to positive infinity, which corresponds with answer choice. C 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. 6x-(2x+3)≥4x-5

Key Concepts

Inequalities

Interval Notation

Algebraic Manipulation

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