Question

Please help 50 points

Answer 1

Explanation:

look at where the dotted line is : it is vertical at x = -3.

that is automatically is equation : x = -3.

now, look at where the shaded side of that line is - to the left or to the right of the line ?

it is to the left of the line. and that message all values smaller than x = -3. but not equal indicated by the dotted line (that means the line itself is not included).

so, this is described by x < -3.

therefore, the first answer option is out.

now, look at the solid line.

what is the y-intercept (the y- value when x = 0) ?

when x = 0, the y-value is -1.

and everything to the left and below of the line is included (incl. the line itself, hence the solid line), so, the line inequality is then (just given by the available choices)

y <= -x - 1

and the 3rd answer option is therefore correct.

Answer 2

`\begin{cases}x < -3\\y \leq -x-1\end{cases}`

Explanation:

When graphing inequalities:

• < or > : dashed line.
• ≤ or ≥ : solid line.
• < or ≤ : shade under the line.
• > or ≥ : shade above the line.

From inspection of the given graph, there is a dashed line at x = -3 with shading under the line (to the left).

Therefore, the inequality that represents this is:

• 
  `x < -3`

The solid line has a negative slope. For each decrease of 1 unit in the y-direction, the x-values increase by 1 unit. Therefore, the slope of this line is -1.

From inspection of the given graph, the y-intercept of the solid line is -1.

Therefore, the equation of this line is y = -x - 1.

As the shading is under the line, the inequality that represents this is:

• 
  `y \leq -x - 1`.

Therefore, the system of linear inequalities that represents the graph is:

`\begin{cases}x < -3\\y \leq -x-1\end{cases}`