1. Do Now 1/25/11 Take out HW from last night. Copy HW in your planner.
Mid-Term Review worksheet #1
Mid-Term Review worksheet #2, #1-13 all
Copy HW in your planner.
Mid-Term tomorrow!
Go to bed by 8:00
Eat breakfast in the morning

2. Algebra Mid-Term Review Worksheet #1
1) -162
2) 3x² - 5y + 1
3) 10
4) x = b – (c/a)
5) -2
6) -1
7) y = 8
8) x = -6
9) no solution
10) no solution
11) vertical
12) y = -5x – 22
13) -15, 9
14) 2 < z < 5
15) undefined / 0
16) x < 1.6
17) x > 9 or x < 9
18) y ≤ 2 and y ≥ -3
19) x > -1 or x < -3
20) -1
21) whole, rational, integer, real
22) 75, 80, 85
23) m = -5/2; y-int = 5
24) y – 2 = 7(x + 4)
25) t ≤ -2
26) empty set / no solution; x = 4
27) 2x + y = 0
28) identity/ all real numbers
29) B
30)

3. Objective
SWBAT review Chapters 4-7 concepts for Mid-Term

4. Chapter 4 Graphing Linear Equations and Functions

5. Section 4.1 “Coordinate Plane”
y-axis
Quadrant II
(-,+)
Quadrant I
(+,+)
Origin
(0,0)
x-axis
Quadrant III
(-,-)
Quadrant IV
(+,-)

6. Section 4.2 “Graph Linear Equations”
Graph the equation y + 2x = 4.
SOLUTION
STEP 2
STEP 3
STEP 1
Solve the equation for y.
Make a table by choosing a few values for x and then finding values for y.
Plot the points. Notice the points appear on a line. Connect the points drawing a line through them. x -2 -1 1 2 y 8 6 4

7. Section 4.3 “Graph Using Intercepts”
Graph the equation 6x + 7y = 42.
Find the x-intercept
Find the y-intercept
6x + 7y = 42
6x + 7y = 42
6x + 7(0)= 42
6(0) + 7y = 42
x = 
x-intercept 7
y = 
y-intercept 6
Plot points. The x-intercept is 7, so plot the point (7, 0). The y- intercept is 6, so plot the point (0, 6). Draw a line through the points.

8. Section 4.4 “Find Slope and Rate of Change”
Find the slope of the line that passes through the points
(0, 6) and (5, –4)
Let (x1, y1) = (0, 6) and (x2, y2) = (5, – 4).
m =
y2 – y1
x2 – x1
Write formula for slope.
– 4 – 6
5 – 0 =
Substitute. 10 5
= – =
– 2
Simplify.

9. Section 4.5 “Graph Using Slope-Intercept Form”
a linear equation written in the form
y-coordinate
x-coordinate
y = mx + b
slope
y-intercept

10. Line a and line c have the same slope, so they are parallel.
Determine which of the lines are parallel.
Find the slope
of each line.
– 1 – 0
– 1 – 2 =
– 1
– 3 1 3 =
Line a: m =
– 3 – (–1 )
0 – 5
– 2
– 5 = 2 5 =
Line b: m =
– 5 – (–3)
– 2 – 4
– 2
– 6 = 1 3 =
Line c: m =
Line a and line c have the same slope, so they are parallel.

11. Section 4.7 “Graph Linear Functions”
Function Notation-
a linear function written in the form y = mx + b where y is written as a function f.
x-coordinate
f(x) = mx + b
This is read
as ‘f of x’
slope
y-intercept
f(x) is another name for y.
It means “the value of f at x.”
g(x) or h(x) can also be used to name functions

12. Graph the Function f(x) = 2x – 3
Graph a Function
Graph the Function f(x) = 2x – 3
SOLUTION
STEP 2
STEP 3
STEP 1
Plot the points. Notice the points appear on a line. Connect the points drawing a line through them.
The domain and range are not restricted therefore, you do not have to identify.
Make a table by choosing a few values for x and then finding values for y. x -2 -1 1 2
f(x) -7 -5 -3

13. Compare graphs with the graph f(x) = x.
Graph the function g(x) = x + 3, then compare it to the parent function f(x) = x.
f(x) = x
g(x) = x + 3
g(x) = x + 3
f(x) = x x
f(x) -5 -2 1 3 x
f(x) -5 -2 1 3 4 6
The graphs of g(x) and f(x) have the same slope of 1.

14. Chapter 5 Writing Linear Equations

15. Section 5.1 & 5.2 “Write Equations in Slope-Intercept Form”
a linear equation written in the form
y-coordinate
x-coordinate
y = mx + b
slope
y-intercept

16. Section 5.3 “Write Linear Equations in Point-Slope Form”
of a linear equation is written as: y
run
y-coordinate
point 1
x-coordinate
point 1
rise
slope x

17. Section 5.4 “Write Linear Equations in Standard Form” (General Form)
The STANDARD FORM of a linear equation is represented as
Ax + By = C
where A, B, and C are real numbers

18. Section 5.5 “Write Equations of Parallel and Perpendicular Lines”
PARALLEL LINES
If two nonvertical lines in the same plane have the same slope, then they are parallel.
If two nonvertical lines in the same plane are parallel, then they have the same slope.

19. Write an equation of the line that passes through (–2,11) and is parallel to the line y = -x + 5.
STEP 1
Identify the slope. The graph of the given equation has a slope of -1. So, the parallel line through (– 2, 11) has a slope of -1.
STEP 2
Find the y-intercept. Use the slope and the given point.
y = mx + b
Write slope-intercept form.
11 = -1(–2) + b
Substitute -1 for m, -2 for x, and 11 for y.
9 = b
Solve for b.
STEP 3
Write an equation. Use y = mx + b.
y = -x + 9
Substitute -1 for m and 9 for b.

20. Section 5.5 “Write Equations of Parallel and Perpendicular Lines”
If two nonvertical lines in the same plane have slopes that are negative reciprocals, then the lines are perpendicular.
If two nonvertical lines in the same plane are perpendicular, then their slopes are negative reciprocals
½ and -2 are negative reciprocals.
3 and -1/3 are negative reciprocals.

21. Using a Line of Fit to Model Data
Modeling Data
When data show a positive or negative correlation, you can model the trend in the data using a
LINE OF FIT
Using a Line of Fit to Model Data .
Make a scatter plot of the data.
Decide whether the data can be a modeled by a line. (Does it have positive or negative correlation?)
Draw a line that appears to fit the data closely.
Write an equation using two points on the line. (The points do not have to be actual data pairs, but they do have to be on the line.)

22. Ounces of water in water bottle
Draw a line of fit for the scatter plot. Write an equation that models the number of ounces of water left in the water bottle as a function of minutes on the treadmill.
Minutes on treadmill 5 10 15 20 25 30 35
Ounces of water in water bottle 12 9 7 3
Write an equation using two points on the line.
y-axis 12
Use the points (5,12) and (30,4).
12 – 4 = _8_
5 – 8
Ounces of water in water bottle
Find the y-intercept. Use (5,12). 4
y = mx + b
12 = 8 (5) + b b = 68 5 10 15 20 25 30 35
x-axis
y = x + 68
Minutes on the treadmill

23. Chapter 6 Solving and Graphing Linear Inequalities

24. Writing Equations with Inequalities
Symbol
Meaning
Key phrases =
Is equal to
The same as
<
Is less than
Fewer than ≤
Is less than or equal to
At most, no more than
>
Is great than
More than ≥
Is greater than or equal to
At least, no less than

25. On a number line, the GRAPH OF AN INEQUALITY in one variable is the set of points that represent all solutions of the inequality.
“Less than” and “greater
than” are represented
with an open circle.
Graph x < 3 -2 -1 1 2 3
“Less than or equal to”
and “greater than or equal
to” are represented with
closed circle.
Graph x ≥ -1 -2 -1 1 2 3

26. Solve Inequalities When Multiplying and Dividing by a NEGATIVE”
Multiplying and/or dividing each side of an inequality by a NEGATIVE number only produces an equivalent inequality IF the inequality sign is REVERSED!!

27. There are no solutions because 5 < – 21 is false. 5 < – 21
Solve: 14x + 5 < 7(2x – 3)
14x + 5 < 7(2x – 3)
Write original inequality.
14x + 5 < 14x – 21
Distributive property
There are no solutions because 5 < – 21 is false.
5 < – 21
Subtract 14x from each side.
**HINT**
If an inequality is equivalent to an inequality that is false,
such as 5 < -21, then the solution of the inequality has
NO SOLUTION.

28. All real numbers are solutions because – 1 > – 6 is true.
12x – 1 > 6(2x – 1)
12x – 1 > 6(2x – 1)
Write original inequality.
12x – 1 > 12x – 6
Distributive property
All real numbers are solutions because – 1 > – 6 is true.
– 1 > – 6
Subtract 12x from each side.
**HINT**
If an inequality is equivalent to an inequality that is true,
such as -1 > -6, then the solutions of the inequality are
ALL REAL NUMBERS .

29. Solve a Compound Inequality with AND
–7 < x – 5 < -1. Graph your solution.
Separate the compound inequality into two inequalities. Then solve each inequality separately.
–7 < x – 5
and
x – 5 < -1
Write two inequalities.
–7 + 5 < x –5 + 5
and
x – <
Add 5 to each side.
– – –
Graph:
–2 < x
and
x < 4
Simplify.
The compound inequality can be written as – 2 < x < 4.

30. Solve a Compound Inequality with OR
Solve 2x + 3 < 9 or 3x – 6 > 12. Graph your solution.
Solve the two inequalities separately.
2x + 3 < 9 or
3x – 6 > 12
Write original
inequality.
2x + 3 – 3 < 9 – 3 or
3x – >
Addition or Subtraction
property of inequality
2x < 6 or
3x > 18
Simplify.
The solutions are all real numbers less than 3 or greater
than 6.
x < 3 or
x > 6
Simplify.

31. Solving an Absolute Value Equation
The equation |ax + b| = c where c ≥ 0,
is equivalent to the statement:
ax + b = c or ax + b = -c

32. Solve an Absolute Value Equation
Solve 4|t + 9| - 5 = 19
First, rewrite the equation in the form ax + b = c.
4 t + 9 – 5 = 19
Write original equation.
4 t + 9 = 24
Add 5 to each side.
t + 9 = 6
Divide each side by 4.
Next, solve the absolute value equation.
t + 9 = 6
Write absolute value equation.
t + 9 = 6 or t + 9 = –6
Rewrite as two equations.
t = –3 or t = –15
Addition & subtraction to each side

33. Solving an Absolute Value Inequality
The inequality |ax + b| < c where c > 0, will
result in a compound AND inequality.
-c < ax + b < c
The inequality |ax + b| > c where c > 0, will
result in a compound OR inequality.
ax + b > c or ax + b < -c

34. Solve an Absolute Value Inequality
Solve |x - 5| > 7. Graph your solution
|x - 5| > 7
Write original inequality.
Rewrite as compound
inequality.
x - 5 > 7
x - 5 < -7 OR +5 +5
Add 5 to each side.
x > OR x < -2
Simplify.
Graph of x > 12 or x < -2 -2 2 4 6 8 10 12

35. Solve an Absolute Value Inequality
Solve |-4x - 5| +3 < 9. Graph your solution
|-4x - 5| +3 < 9
Write original inequality. -3 -3
Subtract 3 from each side.
|-4x - 5| < 6
Rewrite as compound
inequality.
-4x – 5 < 6
-4x - 5 > -6
AND +5 +5
Add 5 to both sides.
-4x < 11
-4x > -1
Divide by -4 to each side.
AND
x > AND x < 0.25
Simplify.
Graph of < x < 0.25 -3 -2 -1 1 2 3

36. Graphing Linear Inequalities
Graphing Boundary Lines:
Use a dashed line for < or >.
Use a solid line for ≤ or ≥.

37. Graph the inequality y > 4x - 3.
Graph an Inequality
Graph the inequality y > 4x - 3.
STEP 2
STEP 3
STEP 1
Graph the equation
Test (0,0) in the original inequality.
Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality.

38. Chapter 7 Systems of Equations and Inequalities

39. “Solve Linear Systems by Substituting”
y = 3x + 2
Equation 1
x + 2y = 11
Equation 2
x + 2(3x + 2) = 11
x + 2y = 11
Substitute
x + 6x + 4 = 11
7x + 4 = 11
x = 1
y = 3x + 2
Equation 1
Substitute value for
x into the original equation
y = 3(1) + 2
y = 5
(5) = 3(1) + 2
5 = 5
The solution is the point (1,5).
Substitute (1,5) into both equations to check.
(1) + 2(5) = 11
11 = 11

40. Section 7.3 “Solve Linear Systems by Adding or Subtracting”
ELIMINATION-
adding or subtracting equations to obtain a new equation in one variable.
Solving Linear Systems Using Elimination
(1) Add or Subtract the equations to eliminate one variable.
(2) Solve the resulting equation for the other variable.
(3) Substitute in either original equation to find the value of the eliminated variable.

41. + “Solve Linear Systems by Elimination Multiplying First!!”
Eliminated
x (2)
4x + 5y = 35
8x + 10y = 70
Equation 1 +
x (-5)
15x - 10y = 45
-3x + 2y = -9
Equation 2
23x = 115
x = 5
4x + 5y = 35
Equation 1
Substitute value for
x into either of the original equations
4(5) + 5y = 35
20 + 5y = 35
y = 3
4(5) + 5(3) = 35
35 = 35
The solution is the point (5,3).
Substitute (5,3) into both equations to check.
-3(5) + 2(3) = -9
-9 = -9

42. Section 7.5 “Solve Special Types of Linear Systems”
consists of two or more linear equations in the same variables.
Types of solutions:
(1) a single point of intersection – intersecting lines
(2) no solution – parallel lines
(3) infinitely many solutions – when two equations represent the same line

43. “Solve Linear Systems with No Solution”
Eliminated
Eliminated
3x + 2y = 10
Equation 1 _ +
-3x + (-2y) = -2
3x + 2y = 2
Equation 2
This is a false statement,
therefore the system has no
solution.
0 = 8
“Inconsistent
System”
No Solution
By looking at the graph, the lines
are PARALLEL and therefore will
never intersect.

44. “Solve Linear Systems with Infinitely Many Solutions”
Equation 1
x – 2y = -4
Equation 2
y = ½x + 2
Use ‘Substitution’ because we know what y is equals.
Equation 1
x – 2y = -4
x – 2(½x + 2) = -4
x – x – 4 = -4
This is a true statement,
therefore the system has infinitely many solutions.
-4 = -4
“Consistent
Dependent
System”
Infinitely Many Solutions
By looking at the graph, the lines are the SAME and therefore intersect at every point, INFINITELY!

45. Section 7.6 “Solve Systems of Linear Inequalities”
SYSTEM OF INEQUALITIES-
consists of two or more linear inequalities in the same variables.
x – y > 7
Inequality 1
2x + y < 8
Inequality 2
A solution to a system of inequalities is an ordered pair (a point) that is a solution to both linear inequalities.

46. Graph a System of Inequalities
y < x – 4
Inequality 1
y ≥ -x + 3
Inequality 2
Graph both inequalities in the same coordinate plane. The graph of the system is the intersection of the two half-planes, which is shown as the darker shade of blue.
(5,0)
0 < 5 – 4 ?
0 ≥ ?
0 < 1
0 ≥ -2

47. Graph a System of THREE Inequalities
Inequality 1:
Graph all three inequalities in the same coordinate plane. The graph of the system is the triangular region, which is shown as the darker shade of blue.
x > -2
Inequality 2:
x + 2y ≤ 4
Inequality 3:
x + 2y ≤ 4 ?
y ≥ -1 ?
x > -2 ?
Check
(0,0)
0 ≥ -1
0 > -2
0 + 0 ≤ 4

48. With your 3:00 partner, complete Mid-Term Review worksheet #2
Clock Partners
With your 3:00 partner, complete Mid-Term Review worksheet #2