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Analysis of the Questions

4. Solve and check inequality $x + y < 7$

The test point provided is $(2, -2)$.

1. Substitute $x = 2$ and $y = -2$ into the inequality: $2 + (-2) < 7 \implies 0 < 7$ This is true, so $(2, -2)$ satisfies the inequality.

5. Solve and check inequality $3x - 4y > 6$

The test point provided is $(-1, -1)$.

1. Substitute $x = -1$ and $y = -1$ into the inequality: $3(-1) - 4(-1) > 6 \implies -3 + 4 > 6 \implies 1 > 6$ This is false, so $(-1, -1)$ does not satisfy the inequality.

6. Solve and check inequality $-x - 2y \geq 5$

The test point provided is $(-2, -3)$.

1. Substitute $x = -2$ and $y = -3$ into the inequality: $-(-2) - 2(-3) \geq 5 \implies 2 + 6 \geq 5 \implies 8 \geq 5$ This is true, so $(-2, -3)$ satisfies the inequality.

Graphing Inequalities (7–12)

8. $y > -1$

1. The boundary line is $y = -1$, a horizontal line at $y = -1$.
2. The inequality $y > -1$ means the solution is the region above the line. The line itself is dashed because the inequality does not include equality.

10. $x \leq -1$

1. The boundary line is $x = -1$, a vertical line at $x = -1$.
2. The inequality $x \leq -1$ means the solution is the region to the left of the line. The line is solid because the inequality includes equality.

Additional Problems

If you want help with problems $7, 9, 11,$ or $12$, let me know!

Follow-up Questions:

1. Would you like a detailed explanation for graphing inequalities on a coordinate plane?
2. Do you need additional help with any specific inequality?
3. Should I explain why we shade specific regions for inequalities?
4. Do you need more examples of substituting test points into inequalities?
5. Would you like assistance with solving compound inequalities?

Tip: When graphing inequalities, always start by drawing the boundary line (solid or dashed), then test a point to determine which region to shade!