Linear Inequalities Linear Equations Drive on The Education

Linear Inequalities Linear Equations Drive on The Education Highway

Linear Equations and Graphing 1. Parts of a Coordinate Plane 2. Slope 3. Slope-intercept Form of a Linear Equation 4. Graphing by x- & y-intercepts.

Parts of a coordinate plane.

Click on the correct quadrant numbers. Correct answer = applause. 6 5 Quadrant I II 4 III IV 3 Quadrant I II IV 2 1 -7 -6 -5 -4 -3 -2 Quadrant I II IV -1 1 -1 2 -2 Quadrant -3 -4 -5 -6 Lesson Start I 3 II 4 5 III 6 7 IV

Click on the correct axis names. Correct answer = clapping. 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 1 -1 x-axis 2 -2 -3 -4 -5 -6 Lesson Start x-axis y-axis 3 4 5 6 7 y-axis

Click on the point for the origin. Correct answer = clapping. 6 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 -5 -6 Lesson Start y 2 3 4 5 6 7

Click on point (-3, 5). Correct point = applause. 6 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 -5 -6 Lesson Start y 2 3 4 5 6 7

You chose a line segment instead of a point. Go back and try again.

You chose point (5, -3). Each ordered pair is in the form (x, y) -- it follows the alphabet in its internal order. You find the x value first, then you find the y value. Where they meet is the point. Go back and try again.

Click on the correct ordered pair for the black point. 6 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 4 5 6 7 -5/4 (-5, 4) -3 (4, -5) -4/5 -5 -6 y 3 -2 -4 Lesson Start 2

You did not choose an ordered pair. Go back and try again.

You chose the ordered pair for the pale green point. Remember: x comes before y in the alphabet and in an ordered pair. Go back and try again.

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Slopes 1. What is a slope? 2. Slope formula 3. Types of slopes

What is slope? Slope is the slant of a line. Slope = rise change in y’s run change in x’s Slope is a fraction/integer. Lesson Start

How to determine the slope when the line goes up. 1. Count the number of units up from the right point to the left point. -7 2. Put that number on top of the fraction line. 5 4 6 1 5 4 -6 3 -5 2 1 2 3 4 5 36 7 8 9 2 1 x -4 -3 -2 -1 1 -1 -2 -3 2 3 4 5 6 7 Slope = 6 9 -4 3, Count the number of units to the y 4. Put that number under the right. fraction line. Lesson Start

How to determine the slope when the line goes down. 1. Count the number of units down from right point to left point. 2. Put that number on top of fraction line. -7 5 4 -1 -2 -6 -5 -4 3 2 -3 -4 -3 -2 -1 -5 -6 1 2 3 1 x 1 -1 -2 -3 2 3 4 5 6 7 Slope = -6 3 -4 3. Count the units to the right. Lesson Start y 4. Put that number under the fraction line.

Determine the slope of the line shown. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 2 3 4 5 -2 -3 -4 y Lesson Start -1/3 3/1 -3/1 1/3 6 7

The line does not go down. Go back and try again. Lesson Start

The line does not rise 3 units, then run 1 unit to the right. Go back and try again. Lesson Start

Determine the slope of the line shown. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 2 3 4 -2 -3 -4 y Lesson Start -2/3 3/2 -3/2 2/3 5 6 7

The line does not go up. Go back and try again. Lesson Start

The line does not rise -2 units, then run 3 units to the right. Go back and try again. Lesson Start

Slope Formula: m = (y 1 - y 2) (x 1 - x 2) where m = slope and (x 1, y 1), (x 2, y 2) are points on the line. Lesson Start

Example: Find the slope for a line with points (-3, 4) and (7, -2) on it. 1. Assign values as follows: (x 1, y 1) = (-3, 4) (x 2, y 2) = (7, -2) 2. Substitute them into the formula and solve. m = 4 - (-2) = 6 = -3 -3 - 7 -10 5 Lesson Start

Find the slope of the line with points (5, 6) and (2, 9) on it. 1. (x 1, y 1) = (5, 6) (2, 9) Lesson Start

Find the slope of the line with points (5, 6) and (2, 9) on it. 1. (x 1, y 1) = (5, 6) 2. (x 2, y 2) = (5, 6) (2, 9) Lesson Start

Find the slope of the line with points (5, 6) and (2, 9) on it. 1. (x 1, y 1) = (5, 6) 2. (x 2, y 2) = (2, 9) 3. m = 6 + 9 5+2 6 -9 5 -2 6 -9 5+2 6+9 Lesson Start

The slope formula is a case of subtraction on top and bottom. Go back and try again. Lesson Start

You have your x’s and y’s upside down. Remember: “Y’s guys are always in the skies!” Go back and try again. Lesson Start

You have your x’s and y’s upside down. You are also adding when you need to subtract. Go back and try again. Lesson Start

Find the slope of the line with points (5, 6) and (2, 9) on it. 1. (x 1, y 1) = (5, 6) 2. (x 2, y 2) = (2, 9) 3. m = 6 - 9 = -3 = -1 5 -2 3 Lesson Start

Find the slope of the line with points (7, 5) and (3, -4) on it. m= 5 -4=1 7 -3 4 7 -3 =4 5 - (-4) 9 5 - (-4) = 9 3 - 7 -4 5 - (-4) = 9 7 -3 4 Lesson Start

You have your x’s and y’s upside down. Remember: “Y’s guys are always in the skies!” Go back and try again. Lesson Start

It is not 5 - 4, it is 5 - (-4). Go back and try again. Lesson Start

You must start with the y and x from the first point and end with the y and x from the second point. Go back and try again. Lesson Start

Types of Slopes: 1. Positive and Negative Slopes 2. Special Types of Slopes 3. Determining Types of Slopes by Looking at Graphs of Lines 4. Determining Types of Slopes Algebraically. Lesson Start

Positive and Negative Slopes. Type Graph Algebra Positive Up left to right. Positive Fraction Negative Down left to right. Negative Fraction Lesson Start

2 Special Types of Slopes Type Graphs Zero. Horizontal Line Undefined Vertical Line No Slope Lesson Start Algebra 0/a, a 0 a/0

Determining Types of Slopes by Looking at Graphs of Lines Lesson Start

Is the slope of the line positive, negative, zero, or undefined? 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 + 0 Lesson Start - y 2 3 4 5 6 7

Is the slope of the line positive, negative, zero, or undefined? 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 + Lesson Start 0 - y 2 3 4 5 6 7

Is the slope of the line positive, negative, zero, or undefined? 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 + 0 Lesson Start - y 2 3 4 5 6 7

Is the slope of the line positive, negative, zero, or undefined? 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 + 0 Lesson Start - y 2 3 4 5 6 7

Click on the line with the negative slope. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

Click on the line with the zero slope. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

Click on the line with the no slope. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

Click on the line with the positive slope. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

Determining Types of Slopes Algebraically. Lesson Start

Is the slope of the line with the 2 points listed positive, negative, zero, or undefined? Points (-3, 5) and (-9, -4) + Lesson Start - 0

Is the slope of the line with the 2 points listed positive, negative, zero, or undefined? Points (3, 5) and (3, -4) + Lesson Start - 0

Is the slope of the line with the 2 points listed positive, negative, zero, or undefined? Points (-3, -5) and (-9, -4) + Lesson Start - 0

Is the slope of the line with the 2 points listed positive, negative, zero, or undefined? Points (-3, -4) and (-9, -4) + Lesson Start - 0

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Slope-intercept Form of a Linear Equation 1. Slope-intercept equation 2. Graphing by slope-intercept 3. Writing slope-intercept equations

Slope-intercept Form: y = mx + b where m = slope and b = y-intercept. Lesson Start

Example: y = -1/2 x + 4 -1/2 = m = slope 4 = b = y-intercept Lesson Start

Graphing by Slope-intercept 1. Graphing lines with positive/negative slopes. 2. Graphing lines with zero or undefined/no slopes. Lesson Start

The Slope-intercept Song You make the last number first. It’s either up or down. Make the slope in 2 numbers, Or you look like a clown. Top one’s up or down, And the bottom’s always right. You’d better do it well, Or you’ll get a fright. Lesson Start (Tune: “Hokey-Pokey”)

Graph y = -1/2 x + 4 1. Last number is 4, so go up 4 on the yaxis from the origin and plot a point. 2. Slope is already 2 numbers. Top one is -1, so go down 1 from the point you just plotted. -7 -6 -5 -4 -3 -2 4 3 3 2 2 1 1 -1 x 1 -1 3. The bottom number is 2, so go 2 units to the right and plot a point. -2 -3 -4 y Lesson Start 4. Draw a line through the 2 points you plotted. 5 2 3 4 5 6 7

Graph y = 2 x - 3 1 1. Last number is -3, so go down 3 units from the origin and plot a point. 2. The slope is only 1 number so put a 1 under the 2. 4. Go 1 unit to the right and plot a point. 5 4 3 2 1 x -7 -6 -5 -4 3. Go up 2 from the point you just plotted. -3 -2 -1 1 2 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7 5. Draw a line through the 2 points you plotted.

Click on the graph for y = 2/3 x - 2 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

The slope is not negative. Go back and try again. Lesson Start

You graphed the last number on the x-axis instead of the y-axis. Go back and try again. Lesson Start

Top number is 2, and the bottom is 3, so you do not go up 3 and over 2. Go back and try again. Lesson Start

Click on the graph for y = -4 x + 3 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

The slope is not positive. Go back and try again. Lesson Start

The -4 is not the y-intercept, nor is the 3 the x-intercept. Go back and try again. Lesson Start

The -4 is the slope, not the x-intercept. Go back and try again. Lesson Start

Two Special Graphs: Line with a zero slope And Line with an undefined slope. Lesson Start

Line with a zero slope: y=# (no x) graphs as a horizontal line. “Why, o y, do I look upon the horizon? ” Lesson Start

Graph the equation y = 2. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

Line with an undefined/no slope: x=# (no y) graphs as a vertical line. Lesson Start

Graph the equation x = -4. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

Click on the graph for x = 3. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

You chose the graph for x = -3. Go back and try again. Lesson Start

The x = # (no y) line does not graph as a horizontal line. Go back and try again. Lesson Start

Click on the graph for y = -3½. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

The y = # (no x) line does not graph as a vertical line. Go back and try again. Lesson Start

You chose the graph for y = 3½. Go back and try again. Lesson Start

Writing Slope-intercept Equations: 1. When given a slope and the y-intercept. m = ¾, b = -1 2. When given a slope and one point on the line. m = -¼, (8, 3) 3. When given 2 points on the line. (3, 7), (5, 12) Lesson Start

Writing a slope-intercept equation when given a slope and the y-intercept. Substitute the slope and the y-intercept for the m and b in the equation. Example: m = ¾, b = -1 y = mx + b Slope-int. equation y = ¾x - 1 The new equation Lesson Start

1. Click on the correct equation for a line with slope = 5/3 and y-intercept = 2. 5/3 y = 2 x + 5/3 y = 5/3 x + 2 y = -5/3 x + 2 2. Click on the correct slope and y-intercept pair for y = 7 x - 5. m = 7, b = -5 Lesson Start m = -5, b = 7 m = -7, b = 5 m = 1/7, b = -5

Writing a slope-intercept equation when given a slope and a point on the line. 1. Substitute the x, y, and m in the slope-intercept form. 2. Solve for b. 3. Substitute the slope and the b in a clean slope-intercept form. Lesson Start

Example: Write the equation of the line with slope = -¼ and point (8, 3). y = mx + b 1. Substitute the slope, x, and y in the equation. 3 = -¼(8) + b 3 = -2 + b +2 +2 5=b y = -¼x + 5 Lesson Start 2. Solve for b. 3. Substitute the slope and b in the equation.

1. Click on the correct substitution for a line with slope = 1/3 and point (5, 9). 9 y = 5 x + 1/3 9 = 1/3 x + 5 9 = 1/3(5) + b 5 = 1/3(9) + b 2. Click on the correct equation for a line with slope = -2/3 and point (-6, 4). y = -2/3 x - 6 Lesson Start y = -2/3 Y = -2/3 x + 4 y = -2/3 x

Writing a slope-intercept equation for a line with 2 points given: 1. Find the slope of the line. 2. Use that slope and the first point to find the y-intercept. 3. Substitute the slope and the y-intercept into the equation. Lesson Start

Example: Write and equation for a line with points (3, 7) & (5, 12). m = (y 1 - y 2) (x 1 - x 2) m = 7 - 12 = -5 = 5 3 -5 Lesson Start -2 1. Find the slope of the line. 2 Continued on next screen.

Write and equation for a line with points (3, 7) & (5, 12). 7 = (5/2)(3) + b m = 5/2 y = mx + b 2(7) = 2(15/2) + 2 b 2. Use the slope and 14 = 15 + 2 b the first point to -15 solve for the -1 = 2 b -1/2 = b 2 y-intercept. 2 Lesson Start Continued on next screen.

Write and equation for a line with points (3, 7) & (5, 12). m = 5/2, b = -1/2 y = mx + b y = 5/2 x - 1/2 Lesson Start 3. Substitute the slope and the y-intercept for the m and the b in the equation.

1. Click on the slope for a line with points (-2, 8) and (7, -5). 3 -9 13 -9 -9 13 13 9 2. Click on the y-intercept for a line with points (-2, 8) and (7, -5). 86 9 Lesson Start -5 86 13 46 9

3. Click on the correct equation for a line with points (3, 7) and (4, 8). y = 3 x + 7 y=x+4 Lesson Start y = 3/4 x + 8 y = -x + 4

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Graphing by x- and y-intercepts. X-intercept: where the line crosses the x-axis. Y-intercept: where the line crosses the y-axis. x-intercept x y-intercept y

How to graph by x- & y-intercepts: 1. Cover the x term with your index finger and solve the resulting equation for y. 2. Go up or down on the y-axis from the origin that many units and plot a point. 3. Cover the y term with your index finger and solve the resulting equation for x. 4. Go left or right on the x-axis from the origin that many units and plot a point. 5. Draw a line through your points. Lesson Start

Graph the line for 3 x + 2 y = 6. 1. Cover the x term and solve for y. 2. Go up 3 units on the y-axis. 3 x + 2 y = 6. y=3 4 3 2 2 1 1 -4 -3 -2 -1 3. Cover-7 the-6 y -5 4. Go right term and solve 2 units on for x. the x-axis. 3 x + 2 y = 6. x=2 1 1 -1 -2 -3 -4 y Lesson Start 5. Draw a line through the points plotted. 5 x 2 3 4 5 6 7

1. Click on the correct intercepts for 3 x - 4 y = 24. x-int: 8 y-int: 6 x-int: 8 y-int: -6 x-int: 6 y-int: 8 x-int: -6 y-int: 8 Lesson Start

2. Click on the graph of 3 x - 6 y = 12. 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 y Lesson Start 2 3 4 5 6 7

3. Click on the correct equation for the line shown. 5 -9 y - 6 x = -36 4 4 x + 6 y = 36 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 -1 6 y + 4 x = 36 -2 -3 -4 y Lesson Start 2 3 4 -6 x - 9 y = -36 5 6 7

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Graphing Linear Inequalities Type of Solving Inequalities Line Where to Shade

How to Determine the Type of Line to Draw Inequality Symbol > or < Type of Line > or < Solid Line Dotted Line

Choose the type of line for the inequality given. 1. y > 3 x - 2 a. Solid b. Dotted 2. y > ¼x - 5 a. Solid Lesson Start b. Dotted

Choose the inequality symbol for the line shown. < or > Lesson Start

Choose the inequality symbol for the line shown. < or > Lesson Start

For Positive, Negative, & Zero Slopes For Undefined or No Slopes

Where to Shade for Positive, Negative, and Zero Slopes: The inequality must be in y mx + b format. can be: >, >, <, or <. Lesson Start

If the Shade inequality is: y > mx + b or Above the line y > mx + b y < mx + b or Below the line y < mx + b Lesson Start

Graph y > x - 2. 1. Graph the line y = x - 2. x 2. Since y >, shade above the line. y Lesson Start

Graph y < x - 2. 1. Graph the line y = x - 2. x 2. Since y <, shade below the line. y Lesson Start

Do you do anything different when the line is dotted rather than solid? Lesson Start

Lesson Start

Graph y > x - 2. 1. Graph the line y = x - 2, but make the line dotted. x 2. Since y >, shade above the line. y Lesson Start

Graph y < x - 2. 1. Graph the line y = x - 2, but make the line dotted. x 2. Since y <, shade below the line. y Lesson Start

Graph y > -½x + 3 Type of line: Solid Dotted x y Lesson Start

Graph y > -½x + 3 Type of line: Solid Dotted x Shade ___ the line. Above Below Lesson Start y

Graph y > -½x + 3 Type of line: Solid Dotted x Shade ___ the line. Above Below Lesson Start y

Choose the correct inequality for the graph shown. y < 1/3 x + 2 x y > 1/3 x + 2 y Lesson Start

Where to Shade for Undefined or No Slopes: The inequality must be in x # (no y) format. can be: >, >, <, or <. Lesson Start

If the inequality is: x># or x># x<# or x<# Lesson Start Shade To the Right of the line Left of the line

Graph x > -2 1. Draw a dotted vertical line at x = -2. x 2. Shade to the right of the line. y Lesson Start

Graph x < -2. 1. Graph the line X = -2. x 2. Shade to the left of the line. y Lesson Start

Graph x > 3. Choose type of line. Solid Dotted x y Lesson Start

Graph x > 3. Choose type of line. Solid x Choose where to shade. Left Right Lesson Start y

Graph x > 3. Choose type of line. Solid x Choose where to shade. Right y Lesson Start

Solving Inequalities You use the same algebraic methods as solving equations except when you multiply/divide both sides by the same negative number. In that case, you switch the direction of the inequality symbol. Lesson Start

Solve -3 x - 2 y < 12 +3 x -2 y < 3 x + 12 -2 -2 -2 y <> -3/2 x - 6 Lesson Start

Choose the correct inequality. 1. 2 x + 5 y > -10 y > 2/5 x + 2 y < -2/5 x - 2 y > -2/5 x - 2 2. 3 x - 2 y > 10 y > -2/3 x - 5 y < 2/3 x - 5 y > 2/3 x - 5 Lesson Start