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# Inequalities with variables on both sides (with parentheses)

## Video transcript

Solve for x. And we have 5x plus 7 is greater than 3 times x plus 1. So let's just try to isolate "x" on one side of this inequality. But before we do that, let's just simplify this righthand side. so we get 5x plus 7 is greater than - let's distribute this 3. So 3 times x plus 1 is the same thing as 3 times x plus 3 times 1 so it's going to be 3x plus 3 times 1 is 3. Now if we want to put our x's on the lefthand side, we can subtract 3x from both sides. That will get rid of this 3x on the righthand side. So let's do that. Let's subtract 3x from both sides, and we get on the lefthand side: 5x minus 3x is 2x plus 7 is greater than - 3x minus 3x - those cancel out. That was the whole point behind subtracting 3x from both sides - is greater than 3. Is greater than 3. No we can subtract 7 from both sides to get rid of this positive 7 right over here. So, let's subtract, let's subtract 7 from both sides. And we get on the lefthand side... 2x plus 7 minus 7 is just 2x. Is greater than 3 minus 7 which is negative 4. And then let's see, we have 2x is greater than negative 4. If we just want an x over here, we can just divide both sides by 2. Since 2 is a positive number, we don't have to swap the inequality. So let's just divide both sides by 2, and we get x is greater than negative 4 divide by 2 is negative 2. So the solution will look like this. Draw the number line. I can draw a straighter number line than that. There we go. Still not that great, but it will serve our purposes. Let's say that's -3, -2, -1, 0, 1, 2, 3. X is greater than negative 2. It does not include negative 2. It is not greater than or equal to negative 2, so we have to exclude negative 2. And we exclude negative 2 by drawing an open circle at negative 2, but all the values greater than that are valid x's that would solve, that would satisfy this inequality. So anything above it - anything above it will work. And let's just try, let's try just try something that should work. and then let's try something that shouldn't work. So 0 should work. It is greater than negative 2. It's right over here. So, let's verify that. 5 times 0 plus 7 should be greater than 3 times 0 plus 1. So this is 7 - 'cause this is just a 0 - 7 should be greater than 3. Right. 3 times 1. So 7 should be greater than 3, and it definitely is. Now let's try something that should not work. Let's try negative 3. So 5 times negative 3... 5 times negative 3 plus 7, let's see if it is greater than 3 times negative 3 plus 1. So this is negative 15 plus 7 is negative 8 That is negative 8. Let's see if that is greater than negative 3 plus 1 is negative 2 times 3 is negative 6. Negative 8 is not - is not greater than negative 6. Negative 8 is more negative than negative 6. It's less than. So, it is good that negative 3 didn't work 'cause we didn't include that in our solution set. So we tried something that is in our solution set and it did work. And something that is not, and it didn't work. So we are feeling pretty good.