Question

Consider the following system of inequalities

y ≤ 9x + 3
y ≤ -8x + 3

Is -1, -6 a solution to the system of inequalities? Explain.
(Dropdown 1: No, yes) because (Dropdown 2: -1, -6 is a solution to only the first equation, -1, -6 is only a solution to the second equation, -1, -6 is a solution to both equations)

Write a graph for the system of inequalities above.

When looking at the graph, where is the solution set to the system of inequalities?
The solution to the inequalities is (Dropdown: Where the shading overlaps, in any of the shaded area)

Could a system of linear inequalities ever have no solutions? What would this look like on the graph?
(Dropdown 1: Yes, no) it would look like (Dropdown 2: Lines that intersect, parallel lines with shading that overlaps, parallel lines with shading that doesn't overlap.)

Answer 1

Dropdown 1: Yes — Dropdown 2: -1, -6 is a solution to both equations.

Work/check:
- For y ≤ 9x + 3: substitute (-1, -6): -6 ≤ 9(−1)+3 = −6 → true (equality).
- For y ≤ −8x + 3: substitute: -6 ≤ −8(−1)+3 = 11 → true.

Graph instructions:
- Graph the two lines y = 9x + 3 and y = −8x + 3 as solid lines (≤ includes the boundary).
- For each line, shade the region below it (because y is ≤ the right-hand side).
- The solution set is the region where both shadings overlap (the common area below both lines).

Dropdown (where solution set): Where the shading overlaps.

Could a system have no solutions?
- Dropdown 1: Yes — Dropdown 2: parallel lines with shading that doesn't overlap.
Explanation: If the lines are parallel and the shaded half-planes are on opposite sides so they do not intersect, there is no common point (no solution). For example y ≥ 2x + 1 and y ≤ 2x − 3.