多步驟的不等式 2 (英)
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多步驟的不等式 2 (英): 多步驟的不等式 2
- 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
- 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.