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Video Transcript (0:06) Linear inequalities are incredibly similar to linear equations in the fact that they relate one variable to another variable. (0:14) Unlike equality, there are actually two kinds of inequality: Strict Weak (0:21) A strict inequality says that something is definitely less than or greater than something else. This is represented by < or >. This is like how a strict parent will make an absolute statement that something is definitely the case! (0:37) For instance, when we say that 3 < 4, we are actually saying that 3 is strictly less than 4. (0:46) We can also use this with variables to state something like x>5, which means that x is strictly greater than 5. (0:54) A weak inequality says that something is less than or equal to or greater than or equal to something else. Weak inequality is represented by ≤ or ≥. This is like how a parent who isn’t strict says that you could do something or you could do something else. (1:14) For instance, I could write x ≤ 9, which tells me that x must be less than or equal to 9, but that is a weaker statement than the strict statement that x<9, which tells me that x would have to be definitely less than 9. (1:30) An easy way to remember all of this notation is that the inequality symbol looks like it wants to eat the number that is larger, just like PacMan! (1:39) Of course, these kinds of statements are a bit boring all by themselves, so it is helpful to relate them to what we know about linear equations. (1:48) Linear inequalities usually take one of two forms: Standard form Slope-intercept form (1:55) The standard form of an inequality is Ax + By ≤ C, but we can replace the less than or equal to sign with any inequality we want! Just like for our linear equations, this form is not particularly useful, but does define a linear inequality. (2:14) The most common form of a linear inequality is the slope-intercept form. This looks like y ≥ mx + b, where m is the slope of the line, b is the y-intercept and x and y are both variables. (2:28) Think about what this inequality is actually saying and suddenly it becomes less cryptic! It states that y must be greater than OR equal to mx + b. This basically means that our y-value will always be on or above the line, we’ll show this idea in greater detail in the lesson on graphing linear inequalities. For now, we’ll have to be satisfied with the concept alone! (2:53) One other key difference between equality and inequality is how to operate using algebra on an inequality. (2:59) For equations, as long as you do the same operation to both sides of the equation, you’ll usually be fine and accurate in your math. (3:07) However, if you have an inequality, there is one operation that you have to be very careful about! Multiplying or dividing by a negative number results in the inequality flipping direction. (3:20) This means that if you have an inequality that looks like -3x ≥ 9, you should divide both sides by -3, but this creates the inequality x ≤ -3. (3:31) Most students know this rule and hate it, since it’s so easy to forget. However, have you ever thought about why on earth we have this rule in the first place? (3:41) What would happen if for -3x ≥ 9 we added 3x to both sides of the inequality? (3:48) We would get 0 ≥ 9+3x. (3:52) Subtract 9 from each side to find -9 ≥ 3x. (3:59) Divide both sides by 3 to find -3 ≥ x. (4:05) Wait a minute… This is the exact same answer we had before and we never flipped the sign! We did way more work, but hopefully you see that dividing (or multiplying) by a negative number is the same as swapping the sides of the inequality. (4:21) This is why we just change the direction of the inequality! Not to confuse students, but to save them all of that effort every time. (4:30) Try to build up your intuition for why we do things in mathematics. When a teacher tells you a rule, don’t follow it blindly; try to understand what you’re actually doing and you won’t forget these basic operations that allow you to solve harder problems.
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Video Transcript (0:05) Graphing linear inequalities combines the skills you already know from graphing linear equations with the skills you’ve gained from understanding the basics of linear inequalities. (0:17) Just like the lesson on linear inequalities, we have to consider whether we have strict (< or >) or weak ( or ) inequality. (0:28) Strict inequality means that we will end up drawing a dashed line to indicate that the line itself is not part of the region. (0:36) Weak inequality means that we will draw a solid line, indicating that the line is part of the region. (0:42) Also, notice how I’ve been saying regions, that’s because the graph of a linear inequality is actually a whole region of our coordinate plane! (0:52) For instance, take the linear inequality x>4. This means that the only restriction we have is that our x value has to be larger than 4 and y can be whatever it wants. This would be graphed with a dashed line (since it is strict inequality). (1:10) Also, notice how on this graph, we see that the line x=4 is a vertical line, which is dashed! This is identical to how you have graphed lines in the past. (1:21) For weak inequality, we can graph a solid line and then determine the region to shade in. (1:27) For instance, take the inequality y2x-3. If we were simply graphing this as an equation, we would be graphing the line y=2x-3 and slope-intercept form tells us that the slope of this line is 2 and it has a y-intercept at -3. (1:49) Now that we have the graph of the line, we must determine what our inequality actually says. Since y is less than or equal to this line, we must shade in the region that is underneath the line! This is graphed in the following way: (2:04) Let’s consider another strict inequality. Take the linear inequality y>-x+2and use the same procedure we outlined above. (2:14) First, graph the line y=-x+2, which has a slope of -1 and a y-intercept of 2. This looks like the following graph. (2:25) Now, consider the region to shade in. Since y is strictly greater than this line, we must change our line into a dashed line and shade the region above it! (2:38) Just like this, we have our linear inequality totally graphed. It’s practically identical to graphing linear equations! (2:46) However, the meaning behind the regions of these linear inequalities is a little unclear. It’s fairly intuitive to see that if y is greater than something it needs to be above and if it is less than something, then it’ll be below. What does this actually mean? (3:05) You can imagine these shaded regions as all of the values of x and y that make the inequality true. This means that the region is actually just representing a set of points where our inequality is true! (3:15) For example, consider the last inequality that we graphed. Once we had graphed y>-x+2, we could choose any point in our shaded region and it should satisfy the inequality. Take the point (2, 1) and substitute these values into the inequality. (3:34) This shows us that 1>-(2)+2, which is basically saying that 1>0. Obviously this is true! (3:43) However, notice what happens when we plug in the point (2, 0) that is on the dashed line. This shows us that 0>-(2)+2, which is saying that 0>0, which is not true! (3:57) This is why we drew a dashed line to begin with. If our inequality had been y-x+2, then that point on our line would have actually worked. (4:09) At this point, you should feel comfortable and confident in graphing linear equations and linear inequalities. Be sure to remember the difference between strict and weak inequality for graphing purposes in particular and understand that when you are shading a region, you’re actually defining a set of points that make your inequality true!
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Graphing linear inequalities is similar to graphing linear equations. The same rules apply for determining the slope and y‐intercept. However, the graphs look slightly different because the lines can either be dashed or solid and include shading depending on the sign of the inequality. The rules for graphing linear inequalities are: Because the ISEE is a multiple test, you will not need to graph linear inequalities by hand. However, you will need to know what the graph of a linear inequality will look like by recognizing patterns from linear equations. Linear inequalities with positive slopes will point upwards and those with negative slopes wil point downwards. If you receive an inequality with y and x, but the equation is not in slope-intercept form (y = mx + b), rearrange the equation so "y" is on the left and everything else on the right. For example: 3x + y = 9 can be rearranged as y = -3x + 9 When deciding between two answer choices that look very similar, plug in values for x or y and match the corresponding points to the correct graph. Try the question below for practice: Which inequality matches the graph below? A) y < x2 + 1 B) y > x2 + 1 C) y ≤ x2 + 2 D) y > x2 + 2 We can eliminate C because we need a solid line for the ≤ sign. We can also eliminate choice A because we shade below the line for the < symbol. Now, we are left between answer choices B and D. Notice that +1 is the y-intercept of this graph, which would correspond to choice B. If you would still like to verify this answer plug in x and y values for choices B and D: The points for choice B matches the graph, so B is the correct answer. Practice Problems Answers to Practice Problems A A D B A B
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