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Linear Equations and Inequalities

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Objectives (1.2.1 and 1.2.2) The students will be able to: solve linear equations and inequalities. determine the equation for a line. describe solutions using numbers, symbols, and/or graphs.

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To solve two-step equations, undo the operations by working backwards. Example: Ask yourself: 1.What is the first thing we are doing to x? 2.What is the second thing? Recall the order of operations as you answer these questions. Dividing by 2 Subtracting 3 To undo these steps, do the opposite operations in opposite order.

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1.Draw the river 2.Add 3 to both sides 3.Simplify 4.Clear the fraction - Multiply both sides by 2 5.Simplify 6.Check your answer Use a DO-UNDO chart as a shortcut to answering the questions. In the table, write the opposite operations in the opposite order DO UNDO ÷2 -3 Follow the steps in the undo column to isolate the variable = - 4 x = – 3 = · 2 2 · · 2

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1) Solve 2x - 1 = x = x = -1 2(-1) - 1 = – 1 = -3 1.Draw the river 2.Add 1 to both sides 3.Simplify 4.Divide both sides by 2 5.Simplify 6.Check your answer D U · ÷ 2

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2) Solve 3y – 1 = 8 A. y = 3 B. y = -3 C. y = D. y = Answer: A

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3) Solve A. d = -7 B. d = -19 C. d = -17 D. d = 17 Answer: B

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Sample HSA Problem: Alexis has $345 in her savings account and will deposit $15 each week. Alexis will not withdraw any money from her savings account. After how many weeks will she have $1,035? Answer: 46 weeks

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What if there are variables on both sides of the equation?

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1) Solve: 3x + 2 = 4x - 1 You need to get the variables on one side of the equation. It does not matter which variable you move. Try to move the one that will keep your variable positive.

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1) Solve 3x + 2 = 4x x 2 = x = x 3(3) + 2 = 4(3) = Draw the river 2.Subtract 3x from both sides 3.Simplify 4.Add 1 to both sides 5.Simplify 6.Check your answer

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A. -3 B. C. D. 3 What is the value of x if 3 - 4x = 18 + x? Answer: A

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3) Solve 4 = 7x - 3x 4 = 4x = x 4 = 7(1) - 3(1) 1.Draw the river. – Notice the variables are on the same side! 2. Combine like terms. 3. Divide both sides by Simplify. 5. Check your answer.

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What is the value of x if x = 12x - 3? A. 0 B. 4 C. No solutions D. Infinite solutions Answer: D

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Solving an Inequality: Solve 5z + 16 < 51 A. z < -35 B. z < -7 C. z < 35 D. z < 7 Answer: D

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Solve: -7(x - 3) < -7 1.Draw the river. 2.Distribute Subtract 21 from both sides. 4. Simplify. 5. Divide both sides by Simplify. 7.Check your answer. -7x + 21 < x < x > 4 Note: > WHY is < reversed???

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Which of these graphs represent the solution set of the inequality A B C D Lets look at #14 in your packet: ? Answer: B

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Definition of Slope Definition of SLOPE: the ratio of the vertical and horizontal distances between any two points on a line. Informally you will see some of these: the slope = the rate of change

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Types of Slope POSITIVE NEGATIVE

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Types of Slope ZERO NO slope

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Counting the slope, given a graph Step 1: find 2 points Step 2: up=pos down = neg right=pos left= neg (4, -2) (-2,8) Step 3: Slope = ____ Simplify = ____ 5 -3

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Counting the slope, given a graph No need to go further – Zero slope No need to go further – No slope

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Finding the slope, given 2 points Remember our formula? (2, 3) and (4, -1) 24 3 (x 1, y 1 ) (x 2, y 2 ) (2, 3) and (4, -1) m = ___ = Dont forget to simplify!

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Finding the slope, given 2 points Remember our formula? (3,6) and (3, 4) (x 1, y 1 ) (x 2, y 2 ) (3,6) and (3, 4) m = ___ 0 2 No slope = Remember when 0 is in the denominator

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Slope as a rate of change Sample Problem: In 1990, a company had a profit of $1,300,000. In 1992, the company had a profit of $1,200,000. Find the average rate of change in dollars per year. A rate of change is a ratio like the slope ,300,0001,200,000dollars year ex: miles per hour mi/hr people per year people/year per is represented by the fraction bar / - - = , ,000 dollars/yr = =

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Practice: Positive Zero No Negative What type of slope is this?

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Example #2: What is the slope?

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Determine the slope of the line through the points (2, 0) and (-4, -5).

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Slope-Intercept Form: y=mx+by=mx+b slopey-intercept

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Q: What is the y-intercept? A: the point where the line crosses the y- axis Example: The line crosses the y-axis at -2. So the y-intercept is the point (0,-2).

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If we have the y-intercept and we can calculate the slope, then we can write the equation of the line. y=mx+b Put in slopePut in y-intercept

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Write the equation of the line. 1)Find the y-intercept: b = 1 2)Find the slope: Lets use the points (0,1) and (2,0) to find the slope m = -1/2 3) Write the equation:

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Find the equation of the line. 1)Find the y-intercept: b = 4 2)Find the slope: Lets use the points (0,4) and (-4,0) to find the slope m = 1/1 3) Write the equation:

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Write the equation of the line. 1)Find the y-intercept: b = -3 2)Find the slope: The line has zero slope; m = 0 3) Write the equation: or

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Write the equation of the line. 1)Find the y-intercept: There is none! 2)Find the slope: The slope is undefined. 3) Write the equation: Point to ponder: Why do you think this is the equation form instead of y=?

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Locating the Intercepts without a Graph: Another way to describe the x and y- intercepts is to think about them as coordinate points on the graph. x-intercept: ( ___, 0) y is always 0. y-intercept: ( 0, ___) x is always 0.

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Without graphing, describe what the graph of the equation would look like. Include information about the slope and y-intercept. 1) y = -5x+32) y = -5 Negative slope m = -5 y-intercept (0,3) Zero slope m = 0 y-intercept (0,-5)

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What if you knew 2 points on a line? What extra step would you have to do to write the equation of the line? Answer: First find the slope, then substitute the coordinates of one point for x and y.

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Example: Find the equation of the line that contains these points: (3, -6),(-5, 2) Slope = Now pick a point to use and solve for b. (Do you think it matters which point we use? Why not?) y = -1x - 3 y = mx + b y = -1x + b -6 = -1(3) + b -3 = b

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You are correct !!

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Incorrect. Please try again.

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The End (But lines do not end!!!!!)

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