Linear Equations and Inequalities

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.

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

Use a DO-UNDO chart as a shortcut to answering the questions 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. Draw “the river” Add 3 to both sides Simplify Clear the fraction -Multiply both sides by 2 Check your answer +3 · 2 + 3 + 3 = - 4 x = -8 -4 – 3 = -7 2 · · 2

1) Solve 2x - 1 = -3 + 1 + 1 2x = -2 2 2 x = -1 2(-1) - 1 = -3 D U 1) Solve 2x - 1 = -3 · 2 - 1 + 1 ÷ 2 + 1 + 1 2x = -2 2 2 x = -1 2(-1) - 1 = -3 -2 – 1 = -3 Draw “the river” Add 1 to both sides Simplify Divide both sides by 2 Check your answer

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

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

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

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

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.

1) Solve 3x + 2 = 4x - 1 - 3x - 3x 2 = x - 1 + 1 + 1 3 = x + 1 + 1 3 = x 3(3) + 2 = 4(3) - 1 9 + 2 = 12 - 1 Draw “the river” Subtract 3x from both sides Simplify Add 1 to both sides Check your answer

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

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

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

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

Solve: -7(x - 3) < -7 - 21 - 21 -7x < -28 -7 -7 x > 4 - 21 - 21 -7x < -28 -7 -7 x > 4 Note: > Draw “the river”. Distribute -7. Subtract 21 from both sides. 4. Simplify. 5. Divide both sides by -7. Simplify. Check your answer. WHY is < reversed???

Let’s look at #14 in your packet: Which of these graphs represent the solution set of the inequality ? A -4 -2 2 4 B -4 -2 2 4 . C -4 -2 2 4 D -4 -2 2 4 Answer: B

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

Types of Slope POSITIVE NEGATIVE

Types of Slope ZERO NO slope

Counting the slope, given a graph Step 2: up=pos down = neg right=pos left= neg Step 1: find 2 points -6 (4, -2) (-2,8) + 10 Step 3: Slope = ____ 10 -6 Simplify = ____ 5 -3

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

Finding the slope, given 2 points Remember our formula? 3 -1 2 4 (x1, y1) (x2, y2) (2, 3) and (4, -1) (2, 3) and (4, -1) 4 m = ___ -2 1 = -2 Don’t forget to simplify!

Finding the slope, given 2 points Remember our formula? 6 4 3 3 (x1, y1) (x2, y2) (3,6) and (3, 4) (3,6) and (3, 4) 2 m = ___ No slope = Remember when 0 is in the denominator

Slope as a rate of change A rate of change is a ratio like the slope. ex: miles per hour ► mi/hr per is represented by the fraction bar / people per year ► people/year 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. dollars year 1,300,000 - = 1,200,000 -2 100,000 = 1990 1992 = - 50,000 dollars/yr

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

Example #2: What is the slope? 5 8 - 5 8 8 5 - 8 5

Determine the slope of the line through the points (2, 0) and (-4, -5). 4 7 5 6 6 5 7 4

Slope-Intercept Form: y=mx+b slope y-intercept

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).

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 slope Put in y-intercept

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

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

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

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=?

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. y-intercept: ( 0, ___) x is always 0. x-intercept: ( ___, 0) y is always 0.

Without graphing, describe what the graph of the equation would look like. Include information about the slope and y-intercept. 1) y = -5x+3 2) y = -5 Negative slope m = -5 y-intercept (0,3) Zero slope m = 0 y-intercept (0,-5)

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.

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

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