Solve The Inequality $4p - 5 \leq 25$.$p$ $\square$

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Introduction to Inequalities

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Inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation, but with an inequality symbol instead of an equal sign.

What are Linear Inequalities?

Linear inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. They are written in the form of a linear equation, but with an inequality symbol instead of an equal sign. For example:

• $2x + 3 \leq 5$
• $x - 2 > 3$
• $4p - 5 \leq 25$

Why are Linear Inequalities Important?

Linear inequalities are an essential part of mathematics, and they have numerous applications in real-life situations. They are used to model and solve problems in various fields, such as economics, finance, and engineering. For instance, a company may use linear inequalities to determine the maximum or minimum profit they can make based on the number of units sold.

Solving Linear Inequalities

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Solving linear inequalities involves isolating the variable on one side of the inequality symbol. This can be done using various techniques, such as adding or subtracting the same value to both sides of the inequality, or multiplying or dividing both sides by the same non-zero value.

Adding or Subtracting the Same Value to Both Sides

When adding or subtracting the same value to both sides of an inequality, the inequality symbol remains the same. For example:

• $2x + 3 \leq 5$
• $2x + 3 - 3 \leq 5 - 3$
• $2x \leq 2$

Multiplying or Dividing Both Sides by the Same Non-Zero Value

When multiplying or dividing both sides of an inequality by the same non-zero value, the inequality symbol remains the same if the value is positive, but it is reversed if the value is negative. For example:

• $2x \leq 2$
• $\frac{2x}{2} \leq \frac{2}{2}$
• $x \leq 1$

Solving the Inequality $4p - 5 \leq 25$

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Now, let's solve the inequality $4p - 5 \leq 25$.

Step 1: Add 5 to Both Sides

First, we add 5 to both sides of the inequality to isolate the term with the variable.

$$4p - 5 + 5 \leq 25 + 5$$

$$4p \leq 30$$

Step 2: Divide Both Sides by 4

Next, we divide both sides of the inequality by 4 to solve for the variable.

$$\frac{4p}{4} \leq \frac{30}{4}$$

$$p \leq \frac{30}{4}$$

Step 3: Simplify the Right-Hand Side

Finally, we simplify the right-hand side of the inequality by dividing 30 by 4.

$$p \leq 7.5$$

Therefore, the solution to the inequality $4p - 5 \leq 25$ is $p \leq 7.5$.

Conclusion

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Solving linear inequalities is an essential part of mathematics, and it has numerous applications in real-life situations. By following the steps outlined in this article, you can solve linear inequalities and understand the concept of inequalities in mathematics. Remember to always check your work and verify the solution to ensure that it is correct.

Final Answer

The final answer is $\boxed{7.5}$.

===========================================================

Introduction to Inequalities

---

Inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation, but with an inequality symbol instead of an equal sign.

What are Linear Inequalities?

Linear inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. They are written in the form of a linear equation, but with an inequality symbol instead of an equal sign. For example:

• $2x + 3 \leq 5$
• $x - 2 > 3$
• $4p - 5 \leq 25$

Why are Linear Inequalities Important?

Linear inequalities are an essential part of mathematics, and they have numerous applications in real-life situations. They are used to model and solve problems in various fields, such as economics, finance, and engineering. For instance, a company may use linear inequalities to determine the maximum or minimum profit they can make based on the number of units sold.

Solving Linear Inequalities

---

Solving linear inequalities involves isolating the variable on one side of the inequality symbol. This can be done using various techniques, such as adding or subtracting the same value to both sides of the inequality, or multiplying or dividing both sides by the same non-zero value.

Adding or Subtracting the Same Value to Both Sides

When adding or subtracting the same value to both sides of an inequality, the inequality symbol remains the same. For example:

• $2x + 3 \leq 5$
• $2x + 3 - 3 \leq 5 - 3$
• $2x \leq 2$

Multiplying or Dividing Both Sides by the Same Non-Zero Value

When multiplying or dividing both sides of an inequality by the same non-zero value, the inequality symbol remains the same if the value is positive, but it is reversed if the value is negative. For example:

• $2x \leq 2$
• $\frac{2x}{2} \leq \frac{2}{2}$
• $x \leq 1$

Solving the Inequality $4p - 5 \leq 25$

---

Now, let's solve the inequality $4p - 5 \leq 25$.

Step 1: Add 5 to Both Sides

First, we add 5 to both sides of the inequality to isolate the term with the variable.

$$4p - 5 + 5 \leq 25 + 5$$

$$4p \leq 30$$

Step 2: Divide Both Sides by 4

Next, we divide both sides of the inequality by 4 to solve for the variable.

$$\frac{4p}{4} \leq \frac{30}{4}$$

$$p \leq \frac{30}{4}$$

Step 3: Simplify the Right-Hand Side

Finally, we simplify the right-hand side of the inequality by dividing 30 by 4.

$$p \leq 7.5$$

Therefore, the solution to the inequality $4p - 5 \leq 25$ is $p \leq 7.5$.

Q&A: Solving Linear Inequalities

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

A: A linear equation is an equation that can be written in the form of ax + b = c, where a, b, and c are constants. A linear inequality, on the other hand, is an inequality that can be written in the form of ax + b ≤ c, where a, b, and c are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality symbol. This can be done using various techniques, such as adding or subtracting the same value to both sides of the inequality, or multiplying or dividing both sides by the same non-zero value.

Q: What is the solution to the inequality $2x + 3 \leq 5$?

A: To solve the inequality $2x + 3 \leq 5$, we need to isolate the variable x. We can do this by subtracting 3 from both sides of the inequality.

$$2x + 3 - 3 \leq 5 - 3$$

$$2x \leq 2$$

Next, we divide both sides of the inequality by 2 to solve for x.

$$\frac{2x}{2} \leq \frac{2}{2}$$

$$x \leq 1$$

Therefore, the solution to the inequality $2x + 3 \leq 5$ is $x \leq 1$.

Q: Can I multiply or divide both sides of an inequality by a negative value?

A: No, you cannot multiply or divide both sides of an inequality by a negative value. When you multiply or divide both sides of an inequality by a negative value, the inequality symbol is reversed.

Q: How do I check my work when solving a linear inequality?

A: To check your work when solving a linear inequality, you need to substitute the solution back into the original inequality and verify that it is true. For example, if you solve the inequality $2x + 3 \leq 5$ and get the solution $x \leq 1$, you need to substitute x = 1 back into the original inequality and verify that it is true.

$$2(1) + 3 \leq 5$$

$$2 + 3 \leq 5$$

$$5 \leq 5$$

Since the inequality is true, the solution $x \leq 1$ is correct.

Conclusion

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Solving linear inequalities is an essential part of mathematics, and it has numerous applications in real-life situations. By following the steps outlined in this article, you can solve linear inequalities and understand the concept of inequalities in mathematics. Remember to always check your work and verify the solution to ensure that it is correct.

Final Answer

The final answer is $\boxed{7.5}$.