complete the table using true or false to indicate if the statement applies to each linear inequality
linear inequality | $y 2x - 1$ (with its graph) | $y \geq 2x - 1$ (with its graph)
--- | --- | ---
coordinates along the boundary line are included in the solution. | |
coordinates along the boundary line are not included in the solution. | |
the y - intercept of the boundary line is -1. | |
$(0, -1)$ is a solution to the inequality | |
$(-1, 3)$ is a solution to the inequality | |
$(2, -2)$ is a solution to the inequality | |

Question

complete the table using true or false to indicate if the statement applies to each linear inequality
linear inequality | $y > 2x - 1$ (with its graph) | $y \geq 2x - 1$ (with its graph)
--- | --- | ---
coordinates along the boundary line are included in the solution. |  |
coordinates along the boundary line are not included in the solution. |  |
the y - intercept of the boundary line is -1. |  |
$(0, -1)$ is a solution to the inequality |  |
$(-1, 3)$ is a solution to the inequality |  |
$(2, -2)$ is a solution to the inequality |  |

Accepted Answer

# Explanation:
## Step1: Analyze $ y > 2x - 1 $
- **Boundary Inclusion (Included):** The inequality is $ > $, so the boundary line $ y = 2x - 1 $ is dashed (not included). So "Coordinates along the boundary line are included" is FALSE.
- **Boundary Inclusion (Not Included):** Since the line is dashed, this is TRUE.
- **y - intercept:** For $ y = 2x - 1 $, when $ x = 0 $, $ y = - 1 $. So the y - intercept is - 1, TRUE.
- **$(0,-1)$ in $ y > 2x - 1 $:** Substitute $ x = 0,y=-1 $: $ - 1>2(0)-1=-1 $? $ - 1>-1 $ is FALSE.
- **$(-1,3)$ in $ y > 2x - 1 $:** Substitute $ x=-1,y = 3 $: $ 3>2(-1)-1=-3 $, $ 3>-3 $ is TRUE.
- **$(2,-2)$ in $ y > 2x - 1 $:** Substitute $ x = 2,y=-2 $: $ - 2>2(2)-1 = 3 $? $ - 2>3 $ is FALSE.

## Step2: Analyze $ y\geq2x - 1 $
- **Boundary Inclusion (Included):** The inequality is $ \geq $, so the boundary line $ y = 2x - 1 $ is solid (included). So "Coordinates along the boundary line are included" is TRUE.
- **Boundary Inclusion (Not Included):** Since the line is solid, this is FALSE.
- **y - intercept:** For $ y = 2x - 1 $, when $ x = 0 $, $ y=-1 $. So the y - intercept is - 1, TRUE.
- **$(0,-1)$ in $ y\geq2x - 1 $:** Substitute $ x = 0,y=-1 $: $ - 1\geq2(0)-1=-1 $, $ - 1\geq - 1 $ is TRUE.
- **$(-1,3)$ in $ y\geq2x - 1 $:** Substitute $ x=-1,y = 3 $: $ 3\geq2(-1)-1=-3 $, $ 3\geq - 3 $ is TRUE.
- **$(2,-2)$ in $ y\geq2x - 1 $:** Substitute $ x = 2,y=-2 $: $ - 2\geq2(2)-1 = 3 $? $ - 2\geq3 $ is FALSE.

| Linear Inequality | $ y > 2x - 1 $ | $ y\geq2x - 1 $ |
| --- | --- | --- |
| Coordinates along the boundary line are included in the solution. | FALSE | TRUE |
| Coordinates along the boundary line are NOT included in the solution. | TRUE | FALSE |
| The y - intercept of the boundary line is - 1. | TRUE | TRUE |
| $(0,-1)$ is a solution to the inequality | FALSE | TRUE |
| $(-1,3)$ is a solution to the inequality | TRUE | TRUE |
| $(2,-2)$ is a solution to the inequality | FALSE | FALSE |

# Answer:
The completed table is as shown above.

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze \( y > 2x - 1 \)

Step2: Analyze \( y\geq2x - 1 \)

Answer:

Linear Inequality	\( y > 2x - 1 \)	\( y\geq2x - 1 \)
Coordinates along the boundary line are NOT included in the solution.	TRUE	FALSE
The y - intercept of the boundary line is - 1.	TRUE	TRUE
\((0,-1)\) is a solution to the inequality	FALSE	TRUE
\((-1,3)\) is a solution to the inequality	TRUE	TRUE
\((2,-2)\) is a solution to the inequality	FALSE	FALSE