Solve Each Inequality. Then Graph Each Solution Set On A Number Line.1. $8x - 7 \ \textgreater \ 9$2. $6 + \frac{2}{5}y \leq \frac{1}{3}y$

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. In this article, we will focus on solving and graphing inequalities, specifically on the number line. We will use two examples to illustrate the process: $8x - 7 \ \textgreater \ 9$ and $6 + \frac{2}{5}y \leq \frac{1}{3}y$. By following these steps, you will be able to solve and graph inequalities with ease.

Example 1: Solving the Inequality $8x - 7 \ \textgreater \ 9$

Step 1: Add 7 to Both Sides of the Inequality

To solve the inequality, we need to isolate the variable $x$. The first step is to add 7 to both sides of the inequality. This will help us get rid of the constant term on the left-hand side.

8x - 7 + 7 \  \textgreater \  9 + 7

Step 2: Simplify the Inequality

After adding 7 to both sides, we can simplify the inequality by combining like terms.

8x \  \textgreater \  16

Step 3: Divide Both Sides of the Inequality by 8

Now that we have isolated the variable $x$, we can divide both sides of the inequality by 8 to solve for $x$.

\frac{8x}{8} \  \textgreater \  \frac{16}{8}

Step 4: Simplify the Inequality

After dividing both sides by 8, we can simplify the inequality by canceling out the common factor.

x \  \textgreater \  2

Step 5: Graph the Solution Set on a Number Line

To graph the solution set on a number line, we need to plot a point that represents the value of $x$ that satisfies the inequality. In this case, we can plot a point at $x = 2$.

x \  \textgreater \  2

The solution set is all real numbers greater than 2.

Example 2: Solving the Inequality $6 + \frac{2}{5}y \leq \frac{1}{3}y$

Step 1: Subtract 6 from Both Sides of the Inequality

To solve the inequality, we need to isolate the variable $y$. The first step is to subtract 6 from both sides of the inequality. This will help us get rid of the constant term on the left-hand side.

6 + \frac{2}{5}y - 6 \leq \frac{1}{3}y - 6

Step 2: Simplify the Inequality

After subtracting 6 from both sides, we can simplify the inequality by combining like terms.

\frac{2}{5}y \leq \frac{1}{3}y - 6

Step 3: Add 6 to Both Sides of the Inequality

Now that we have isolated the variable $y$, we can add 6 to both sides of the inequality to get rid of the constant term on the right-hand side.

\frac{2}{5}y + 6 \leq \frac{1}{3}y + 6

Step 4: Subtract $\frac{2}{5}y$ from Both Sides of the Inequality

After adding 6 to both sides, we can subtract $\frac{2}{5}y$ from both sides to isolate the variable $y$.

6 \leq \frac{1}{3}y - \frac{2}{5}y + 6

Step 5: Simplify the Inequality

After subtracting $\frac{2}{5}y$ from both sides, we can simplify the inequality by combining like terms.

6 \leq \frac{5}{15}y + \frac{6}{15}y + 6

Step 6: Combine Like Terms

After combining like terms, we can simplify the inequality further.

6 \leq \frac{11}{15}y + 6

Step 7: Subtract 6 from Both Sides of the Inequality

Now that we have isolated the variable $y$, we can subtract 6 from both sides of the inequality to get rid of the constant term on the right-hand side.

0 \leq \frac{11}{15}y

Step 8: Multiply Both Sides of the Inequality by $\frac{15}{11}$

After subtracting 6 from both sides, we can multiply both sides of the inequality by $\frac{15}{11}$ to solve for $y$.

0 \leq \frac{15}{11} \cdot \frac{11}{15}y

Step 9: Simplify the Inequality

After multiplying both sides by $\frac{15}{11}$, we can simplify the inequality by canceling out the common factor.

0 \leq y

Step 10: Graph the Solution Set on a Number Line

To graph the solution set on a number line, we need to plot a point that represents the value of $y$ that satisfies the inequality. In this case, we can plot a point at $y = 0$.

y \geq 0

The solution set is all real numbers greater than or equal to 0.

Conclusion

In the previous article, we covered the basics of solving and graphing inequalities on a number line. However, we know that practice makes perfect, and there's always room for improvement. In this article, we'll answer some frequently asked questions about solving and graphing inequalities, providing you with a deeper understanding of this important mathematical concept.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, follow these steps:

1. Add, subtract, multiply, or divide both sides of the inequality by the same value to isolate the variable.
2. Simplify the inequality by combining like terms and canceling out common factors.
3. Graph the solution set on a number line.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, follow these steps:

1. Factor the quadratic expression, if possible.
2. Set each factor equal to zero and solve for the variable.
3. Graph the solution set on a number line.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that uses the symbols $<$ or $>$, such as $x < 2$ or $x > 3$. A non-strict inequality, on the other hand, uses the symbols $\leq$ or $\geq$, such as $x \leq 2$ or $x \geq 3$.

Q: How do I graph a solution set on a number line?

A: To graph a solution set on a number line, follow these steps:

1. Plot a point that represents the value of the variable that satisfies the inequality.
2. Draw an open circle around the point, if the inequality is strict.
3. Draw a closed circle around the point, if the inequality is non-strict.
4. Shade the region to the left or right of the point, depending on the direction of the inequality.

Q: What are some common mistakes to avoid when solving and graphing inequalities?

A: Some common mistakes to avoid when solving and graphing inequalities include:

• Not simplifying the inequality by combining like terms and canceling out common factors.
• Not graphing the solution set on a number line.
• Not using the correct symbols for strict and non-strict inequalities.
• Not considering the direction of the inequality when graphing the solution set.

Q: How can I practice solving and graphing inequalities?

A: There are many ways to practice solving and graphing inequalities, including:

• Working through example problems in a textbook or online resource.
• Creating your own example problems and solving them.
• Using online tools or software to practice solving and graphing inequalities.
• Joining a study group or working with a tutor to practice solving and graphing inequalities.

Conclusion

In this article, we've answered some frequently asked questions about solving and graphing inequalities, providing you with a deeper understanding of this important mathematical concept. By following the steps outlined above and practicing regularly, you'll become proficient in solving and graphing inequalities, and you'll be able to apply this skill to a wide range of mathematical problems.