Solve Each Compound Inequality. Then Graph The Solution Set.3. $y - 1 \geq 7$ Or $y + 3 \ \textless \ -1$4. $t + 14 \geq 15$ Or $t - 9 \ \textless \ -10$13. $10m - 7 \ \textless \ 17m$ Or $-6m \

Introduction

In mathematics, inequalities are used to compare the values of two or more expressions. Compound inequalities are a combination of two or more inequalities, connected by logical operators such as "or" or "and". In this article, we will focus on solving compound inequalities and graphing the solution sets.

Solving Compound Inequalities

To solve a compound inequality, we need to solve each inequality separately and then combine the solutions using the logical operator. Let's consider the following compound inequality:

$y - 1 \geq 7$ or $y + 3 \ \textless \ -1$

Step 1: Solve the first inequality

$y - 1 \geq 7$

Add 1 to both sides:

$y \geq 8$

Step 2: Solve the second inequality

$y + 3 \ \textless \ -1$

Subtract 3 from both sides:

$y \ \textless \ -4$

Step 3: Combine the solutions

Since the logical operator is "or", we need to find the union of the two solution sets. The solution set is the set of all values that satisfy either of the two inequalities.

$y \geq 8$ or $y \ \textless \ -4$

The solution set is the union of the two intervals: $(-\infty, -4) \cup [8, \infty)$.

Graphing the Solution Set

To graph the solution set, we need to plot the two intervals on a number line. The first interval, $(-\infty, -4)$, is the set of all values less than -4. The second interval, $[8, \infty)$, is the set of all values greater than or equal to 8.

The graph of the solution set is a union of two intervals, with a gap between them.

$t + 14 \geq 15$ or $t - 9 \ \textless \ -10$

Step 1: Solve the first inequality

$t + 14 \geq 15$

Subtract 14 from both sides:

$t \geq 1$

Step 2: Solve the second inequality

$t - 9 \ \textless \ -10$

Add 9 to both sides:

$t \ \textless \ -1$

Step 3: Combine the solutions

Since the logical operator is "or", we need to find the union of the two solution sets. The solution set is the set of all values that satisfy either of the two inequalities.

$t \geq 1$ or $t \ \textless \ -1$

The solution set is the union of the two intervals: $(-\infty, -1) \cup [1, \infty)$.

Graphing the Solution Set

To graph the solution set, we need to plot the two intervals on a number line. The first interval, $(-\infty, -1)$, is the set of all values less than -1. The second interval, $[1, \infty)$, is the set of all values greater than or equal to 1.

The graph of the solution set is a union of two intervals, with a gap between them.

$10m - 7 \ \textless \ 17m$ or $-6m \ \textless \ 5$

Step 1: Solve the first inequality

$10m - 7 \ \textless \ 17m$

Subtract 10m from both sides:

$-7 \ \textless \ 7m$

Divide both sides by 7:

$-1 \ \textless \ m$

Step 2: Solve the second inequality

$-6m \ \textless \ 5$

Divide both sides by -6:

$m \ \textgreater \ -\frac{5}{6}$

Step 3: Combine the solutions

Since the logical operator is "or", we need to find the union of the two solution sets. The solution set is the set of all values that satisfy either of the two inequalities.

$m \ \textless \ -1$ or $m \ \textgreater \ -\frac{5}{6}$

The solution set is the union of the two intervals: $(-\infty, -1) \cup (-\frac{5}{6}, \infty)$.

Graphing the Solution Set

To graph the solution set, we need to plot the two intervals on a number line. The first interval, $(-\infty, -1)$, is the set of all values less than -1. The second interval, $(-\frac{5}{6}, \infty)$, is the set of all values greater than -\frac{5}{6}.

The graph of the solution set is a union of two intervals, with a gap between them.

Conclusion

In this article, we have learned how to solve compound inequalities and graph the solution sets. We have used the logical operator "or" to combine the solutions of two inequalities. We have also graphed the solution sets using number lines. By following these steps, we can solve compound inequalities and graph the solution sets.

Key Takeaways

• To solve a compound inequality, we need to solve each inequality separately and then combine the solutions using the logical operator.
• The solution set is the set of all values that satisfy either of the two inequalities.
• To graph the solution set, we need to plot the two intervals on a number line.
• The graph of the solution set is a union of two intervals, with a gap between them.

Practice Problems

1. Solve the compound inequality: $x + 2 \geq 10$ or $x - 3 \ \textless \ -5$
2. Solve the compound inequality: $y - 4 \ \textless \ 2$ or $y + 2 \geq 8$
3. Solve the compound inequality: $t + 5 \ \textless \ 12$ or $t - 2 \geq 9$

Answer Key

1. $x \geq 8$ or $x \ \textless \ -3$
2. $y \ \textless \ 6$ or $y \geq 10$
3. $t \ \textless \ 7$ or $t \geq 11$
   Compound Inequality Q&A ==========================

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities, connected by logical operators such as "or" or "and".

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each inequality separately and then combine the solutions using the logical operator.

Q: What is the difference between "or" and "and" in compound inequalities?

A: In compound inequalities, "or" means that the solution set is the union of the two solution sets, while "and" means that the solution set is the intersection of the two solution sets.

Q: How do I graph the solution set of a compound inequality?

A: To graph the solution set of a compound inequality, you need to plot the two intervals on a number line. The graph of the solution set is a union of two intervals, with a gap between them.

Q: Can I use the same method to solve compound inequalities with different logical operators?

A: Yes, you can use the same method to solve compound inequalities with different logical operators. However, you need to adjust the solution set accordingly.

Q: How do I determine the solution set of a compound inequality with "or"?

A: To determine the solution set of a compound inequality with "or", you need to find the union of the two solution sets.

Q: How do I determine the solution set of a compound inequality with "and"?

A: To determine the solution set of a compound inequality with "and", you need to find the intersection of the two solution sets.

Q: Can I use the same method to solve compound inequalities with different types of inequalities?

A: Yes, you can use the same method to solve compound inequalities with different types of inequalities. However, you need to adjust the solution set accordingly.

Q: How do I solve a compound inequality with absolute value?

A: To solve a compound inequality with absolute value, you need to isolate the absolute value expression and then solve the resulting inequality.

Q: Can I use the same method to solve compound inequalities with absolute value and different logical operators?

A: Yes, you can use the same method to solve compound inequalities with absolute value and different logical operators. However, you need to adjust the solution set accordingly.

Q: How do I graph the solution set of a compound inequality with absolute value?

A: To graph the solution set of a compound inequality with absolute value, you need to plot the two intervals on a number line. The graph of the solution set is a union of two intervals, with a gap between them.

Q: Can I use the same method to solve compound inequalities with different types of variables?

A: Yes, you can use the same method to solve compound inequalities with different types of variables. However, you need to adjust the solution set accordingly.

Q: How do I solve a compound inequality with multiple variables?

A: To solve a compound inequality with multiple variables, you need to solve each inequality separately and then combine the solutions using the logical operator.

Q: Can I use the same method to solve compound inequalities with multiple variables and different logical operators?

A: Yes, you can use the same method to solve compound inequalities with multiple variables and different logical operators. However, you need to adjust the solution set accordingly.

Q: How do I graph the solution set of a compound inequality with multiple variables?

A: To graph the solution set of a compound inequality with multiple variables, you need to plot the two intervals on a number line. The graph of the solution set is a union of two intervals, with a gap between them.

Conclusion

In this article, we have answered some common questions about compound inequalities. We have covered topics such as solving compound inequalities, graphing solution sets, and using different logical operators. We have also discussed how to solve compound inequalities with absolute value and multiple variables. By following these steps, you can solve compound inequalities and graph the solution sets.

Key Takeaways

• To solve a compound inequality, you need to solve each inequality separately and then combine the solutions using the logical operator.
• The solution set is the set of all values that satisfy either of the two inequalities.
• To graph the solution set, you need to plot the two intervals on a number line.
• The graph of the solution set is a union of two intervals, with a gap between them.
• You can use the same method to solve compound inequalities with different logical operators, absolute value, and multiple variables.

Practice Problems

1. Solve the compound inequality: $x + 2 \geq 10$ or $x - 3 \ \textless \ -5$
2. Solve the compound inequality: $y - 4 \ \textless \ 2$ or $y + 2 \geq 8$
3. Solve the compound inequality: $t + 5 \ \textless \ 12$ or $t - 2 \geq 9$

Answer Key

1. $x \geq 8$ or $x \ \textless \ -3$
2. $y \ \textless \ 6$ or $y \geq 10$
3. $t \ \textless \ 7$ or $t \geq 11$