Which graph shows the solution to the system of linear inequalities?

y < 2x – 5

y > –3x + 1

On a coordinate plane, 2 straight lines are shown. The first solid line has a negative slope and goes through (0, 1) and (1, negative 2). Everything to the right of the line is shaded. The second dashed line has a positive slope and goes through (0, negative 5) and (2, negative 1). Everything to the right of the line is shaded.
On a coordinate plane, 2 straight lines are shown. The first dashed line has a negative slope and goes through (0, 1) and (1, negative 2). Everything to the right of the line is shaded. The second solid line has a positive slope and goes through (0, negative 5) and (2, negative 1). Everything to the right of the line is shaded.
On a coordinate plane, 2 straight lines are shown. The first solid line has a negative slope and goes through (0, 1) and (1, negative 2). Everything to the left of the line is shaded. The second dashed line has a positive slope and goes through (0, negative 5) and (2, negative 1). Everything to the left of the line is shaded.
On a coordinate plane, 2 straight lines are shown. The first dashed line has a negative slope and goes through (0, 1) and (1, negative 2). Everything to the left of the line is shaded. The second solid line has a positive slope and goes through (0, negative 5) and (2, negative 1). Everything to the left of the line is shaded.

Let's break down the inequalities and match them with the graph descriptions:

1. **First inequality:** \( y < 2x - 5 \)
   - This represents a **dashed** line because the inequality is **strict** (\(<\)).
   - The region **below** the line will be shaded since the inequality is "less than."

2. **Second inequality:** \( y > -3x + 1 \)
   - This represents a **solid** line because the inequality is **non-strict** (\(>\)).
   - The region **above** the line will be shaded since the inequality is "greater than."

Now, let's match the graph descriptions:

- **Solid line** with a positive slope: This corresponds to the second inequality (\( y > -3x + 1 \)).
- **Dashed line** with a steeper slope: This corresponds to the first inequality (\( y < 2x - 5 \)).

The regions where the shading overlaps is the solution to the system of inequalities.

### Analyzing the options:
- The first option describes the solid line on the right and dashed line on the right, which doesn't match the inequality conditions.
- The second option describes the dashed line on the left and the solid line on the right, which seems to match the conditions for shading, but the solid line should go **above**, not the right of the dashed line.
- The third option describes the shading of everything to the **left** of the lines, which contradicts the inequalities that involve shading above and below the lines.
- The fourth option describes the correct placement of the **solid** line (positive slope) and **dashed** line (negative slope), and the shading is done on the correct sides of each line.

**Therefore, the fourth option is the correct answer.**

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Would you like further clarification on any of these steps? Here are some related questions to explore:

1. What does the difference between a solid and dashed line represent in inequalities?
2. How can we determine which side of the line to shade when graphing linear inequalities?
3. How can we solve a system of linear inequalities algebraically?
4. What is the process for graphing a system of inequalities with one dashed and one solid line?
5. How would the graph change if the inequalities were \( y \leq 2x - 5 \) and \( y \geq -3x + 1 \)?

**Tip:** To help visualize linear inequalities, always start by graphing the lines as equations and then determine the shading based on the inequality signs.

Which graph shows the solution to the system of linear inequalities?

y < 2x – 5

y > –3x + 1

Learn how to graph and solve the system of linear inequalities y < 2x - 5 and y > -3x + 1. This problem explores graphing techniques, including how to shade regions based on inequality signs. Suitable for students in Grades 8-10.

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