Algebra

Algebra: Solving Inequalities

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Welcome to the lecture on “Solving Inequality Equations”, or “Algebraic Inequalities” Now we start We have such an inequality: x> 5 x can be 5.01 or 5.5, or 1 million But it cannot be 4, 3, 0 or -8 In order to see more clearly, we draw a number line This is the number line This is 5 Because x cannot be equal to 5, we draw a hollow circle on 5 Then we bold all possible values \u200b\u200bof x x can be 5.000001 As long as it is a little bit smaller than 5, the inequality is satisfied, right? We write a few numbers that satisfy this inequality 6 satisfied, 10 satisfied, 100 satisfied Now, we multiply or divide both sides of the inequality by -1 I want to know how the inequality will change In other words, what is the relationship between -x and -5? Is it -x> -5 or -x <-5 6 satisfies the inequality x> 5 So is -6 larger than -5 or smaller than -5? -6 is less than -5! Right? I’ll draw another number line here -5 at this point We draw a circle on -5, because -x cannot be equal to -5 )×(……×&(%% 6 satisfy this x -6 is here (to the left of -5), right? Write -6 -6 is less than -5, so is -10, so is -100, and so is -1 million, right? It turns out that -x is less than -5 (-x <-5) When solving algebraic inequalities, you only need to remember: You can treat the> and 5, then -x <-5, try to bring in some different numbers of x So you have a good intuition for this transformation Now we make some connections If 3x+2 is less than or equal to 1, This question is very simple Now subtract 2 from both sides of the inequality When adding or subtracting a number from both sides, the direction of the inequality sign remains unchanged So subtract 2 from both sides, and get 3x less than or equal to -1, right? Now, divide both sides by 3 and get x less than or equal to -1/3 We did not change this inequality, because we divided both sides by a positive 3, right? Now, let’s change a little bit of solution If you subtract 1 from both sides, 3x plus 1 is less than or equal to 0, right? I just subtracted 1 on both sides Now I subtract 3x on both sides Get 1 is less than or equal to -3x, right? I just subtract 3x on the left and 3x on the right Now, I divide both sides by a negative number, and I divide both sides by -3 So I got -1/3 on the left From what we have just learned, both sides of the inequality are divided by a negative number at the same time, and the inequality sign changes direction It used to be less than or equal to sign, now it has become greater than or equal to sign, it has become greater than or equal to x We got the same result through two different methods In the method on the left, we get that x is less than or equal to -1/3 In the method on the right, we get that -1/3 is greater than or equal to x, which is the same result: x is less than or equal to -1/3, right? This is the magic of algebra. You can solve equations in a variety of different ways, and you can always get the same answer as long as you make no mistakes. Let’s do a few more questions Now make a slightly more difficult -8x plus 7 is greater than 5x plus 2 We subtract 5x on both sides Get -13x plus 7 which is greater than 2 Now you can subtract 7 from both sides, and get -13x greater than -5 Now I want to divide both sides by -13 very simple! Get x on the left, divide -5 by -13 on the right to get 5/13, and divide negative numbers by negative numbers to get positive Because both sides are divided by a negative number at the same time, the inequality sign reverses the direction, and x is less than 5/13 Similarly, if you don’t believe my solution, try to substitute some numbers in I remember when I first learned inequality, I didn’t believe in the teacher’s solution I did try to substitute some numbers to confirm, and I believed that the teacher’s method was correct That is: both sides of the inequality are multiplied or divided by a negative number at the same time, and the inequality sign changes direction Remember, only when you multiply and divide negative numbers, not when you add and subtract negative numbers I think these will give you a good understanding of how to solve inequality equations Actually, there is not much new content here When you solve an inequality, or inequality equation You are like solving ordinary equations Use the same method to solve ordinary linear equations The only difference is: When multiplying or dividing both sides of an inequality by a negative number, you need to change the direction of the inequality sign I think you should be able to solve inequalities now I hope you have a good time doing the exercises!

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