Guidance for Graphing Linear Inequalities and Systems

Q15. Graph the following inequality: $x - y < 1$

Background

Topic: Linear Inequalities

This question tests your ability to graph a linear inequality in two variables. You need to understand how to represent the solution set for an inequality on the coordinate plane.

Key Terms and Formulas:

• Linear inequality: An inequality involving two variables, such as $x - y < 1$.

• Boundary line: The line $x - y = 1$ is used as the boundary for the inequality.

• Shading: The region that satisfies the inequality is shaded.

Step-by-Step Guidance

1. Rewrite the inequality as an equation to find the boundary line: $x - y = 1$.

2. Determine if the boundary line should be solid or dashed. Since the inequality is "<" (not "≤"), the line should be dashed to indicate points on the line are not included.

3. Find two points to plot the boundary line. For example, when $x = 0$, $y = -1$; when $y = 0$, $x = 1$.

4. Choose a test point (such as $(0,0)$) and substitute into the original inequality to determine which side of the line to shade.

Try solving on your own before revealing the answer!

Final Answer:

The solution is the region below the dashed line $x - y = 1$.

We shade the area where $x - y < 1$ holds true, which is confirmed by testing a point like $(0,0)$.

Q16. Graph the solution of the system of linear inequalities: $x \leq 4$, $y \geq -2$

Background

Topic: Systems of Linear Inequalities

This question tests your ability to graph the solution set for a system of linear inequalities. You need to find the region where both inequalities are satisfied.

Key Terms and Formulas:

• System of inequalities: Two or more inequalities graphed together.

• Boundary lines: $x = 4$ (vertical line), $y = -2$ (horizontal line).

• Shading: The region where both inequalities overlap is the solution.

Step-by-Step Guidance

1. Draw the boundary lines: $x = 4$ (solid, since "≤") and $y = -2$ (solid, since "≥").

2. Shade the region to the left of $x = 4$ for $x \leq 4$.

3. Shade the region above $y = -2$ for $y \geq -2$.

4. The solution is the intersection of the two shaded regions.

Try solving on your own before revealing the answer!

Final Answer:

The solution is the region left of $x = 4$ and above $y = -2$, including the boundary lines.

This region represents all points $(x, y)$ where $x \leq 4$ and $y \geq -2$.

Q17. Graph the solutions of the given system of linear inequalities: $y < 3x - 4$, $y \leq x + 1$

Background

Topic: Systems of Linear Inequalities

This question tests your ability to graph two linear inequalities and find the region where both are satisfied.

Key Terms and Formulas:

• Linear inequalities: $y < 3x - 4$ and $y \leq x + 1$.

• Boundary lines: $y = 3x - 4$ (dashed), $y = x + 1$ (solid).

• Shading: The solution is the region where both inequalities overlap.

Step-by-Step Guidance

1. Draw the boundary lines: $y = 3x - 4$ (dashed, since "<") and $y = x + 1$ (solid, since "≤").

2. Shade below $y = 3x - 4$ for $y < 3x - 4$.

3. Shade below $y = x + 1$ for $y \leq x + 1$.

4. The solution is the region where both shaded areas overlap.

Try solving on your own before revealing the answer!

Final Answer:

The solution is the region below both lines, where $y < 3x - 4$ and $y \leq x + 1$.

This region represents all points $(x, y)$ that satisfy both inequalities.