1. Reasoning with Equations and Inequalities
Unit 2
Reasoning with Equations and Inequalities

2. Solve the following equations: 𝟔𝒙−𝟕=𝒙+𝟑 −𝟑𝒙=𝟏𝟓 𝟐𝒙 𝟑 =𝟒 𝟐𝒙−𝟒=𝟏𝟎
Warm-Up
Solve the following equations: 𝟔𝒙−𝟕=𝒙+𝟑 −𝟑𝒙=𝟏𝟓 𝟐𝒙 𝟑 =𝟒 𝟐𝒙−𝟒=𝟏𝟎

3. Graph Using Slope-Intercept Form
Steps
Solve for slope-intercept form: 𝒚= 𝒎𝒙+𝒃
Graph the y-intercept 𝒃
Use the slope 𝒎 to find the next point
Use the points to draw a line!

4. Slope = rise run slope =𝑚 Positive Slope = UP to the right
Negative Slope = DOWN to the right

5. Graph 𝑦=2𝑥+3
What is your slope?
What is your y- intercept?

6. System of 2 Linear Equations
2 equations with 2 variables (x & y) each
Ax + By = C
Dx + Ey = F
A, B, C, D, E, and F are all numbers
Solution of a System – an ordered pair, (x,y) that makes both equations true

7. Check whether the ordered pairs are solutions of the system:
x - 3y = -5 -2x + 3y = 10
(-5,0)
-5-3(0)= -5
-5 = -5
-2(-5)+3(0)=10
10=10
Solution
(1,4)
1-3(4)= -5
1-12= -5
-11 = -5
*doesn’t work in the 1st equation, no need to check the 2nd
Not a solution

8. Solving Systems by Graphing
Graph each equation on the same coordinate plane.
If the lines intersect: The point (ordered pair) where the lines intersect is the solution.

9. Solving Systems by Graphing
Make sure each equation is in slope- intercept form: 𝑦=𝑚𝑥+𝑏
Graph each equation on the same graph paper.
The point where the lines intersect is the solution. If they don’t intersect then there’s no solution.
Check your solution algebraically!

10. How do you check the solution?
Solve the system graphically: y = 3x – 12 y = -2x + 3
(3, -3)
How do you check the solution?

11. Solving Systems by Graphing
Types of solutions:
If the lines have the same y-intercept 𝑏, and the same slope 𝑚, then the system is dependent. The lines fall on top of each other!
If the lines have the same slope 𝑚, but different y-intercepts 𝑏, then system is inconsistent. Parallel lines have no solutions!
If the lines have different slopes 𝑚, the system is independent.

12. Independent
Inconsistent
Dependent

13. Solving Systems by Elimination
Arrange the equations with like terms in columns.
Multiply, if necessary, to create opposite coefficients for one variable.
Add/Subtract the equations.
Substitute the value to solve for the other variable.
Write your answer as an ordered pair.
Check your answer.

14. 2x – 2y = -8 2x + 2y = 4
2(-1) – 2y = -8
4x = -4
-2– 2y = -8
x = -1
– 2y = -6
y = 3
The solution is (-1,3).

15. Solving Systems by Substitution
Solve an equation for one variable
Isolate 𝑥 or 𝑦
Substitute
Plug what 𝑥 or 𝑦 is into the other equation
Solve the equation
Reverse PEMDAS
Plug back to find the other variable
Plug the value into the equation and solve
Check your solution
Plug the solution back into both equations

16. 2x – 3y = -2
y = -x + 4
𝑦 is already by itself!
2x – 3(-x + 4) = -2
2x + 3x - 12 = -2
5x – 12 = -2
5x = 10
x = 2
y =
y = 2
The solution is (2,2).

17. Word Problems Define variables Write as a system of equations.
Solve showing all steps.
State your solutions in words.

18. Example 1: Work Schedule
You worked 18 hours last week and earned a total of $124 before taxes. Your job as a lifeguard pays $8 per hour, and your job as a cashier pays $6 per hour. How many hours did you work at each job?
x = hours as lifeguard
x + y = 18
8x + 6y = 124
y = hours as cashier

19. Graphing a Linear Inequality
Sketch the line given by the equation (solid if ≥ or ≤, dashed if < or >). This line separates the coordinate plane into 2 halves.
In one half-plane – all of the points are solutions of the inequality.
In the other half-plane - no point is a solution
2. You can decide which half to shade by testing ONE point.
3. Shade the half that has the solutions to the inequality.

20. 𝑦 ≥ − 𝟑 𝟐 𝒙+ 1

21. Graphing Systems of Linear Inequalities
Graph the lines and appropriate shading for each inequality on the same coordinate plane.
Remember: dotted for < and >, and solid for ≥ and ≤.
The solution is the section where all the shadings overlap.

23. Solve the following equations:
Ticket Out the Door
Solve the following equations:
1. 3 𝑥−2 =13
7𝑥−9=3𝑥+11
8−2𝑥=12