Solve The Inequality:$\[ 5x + 2 \leq 17 \\]

Introduction

Inequalities are mathematical expressions that compare two values using a relation such as greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves isolating the variable on one side of the inequality sign and finding the values of the variable that satisfy the inequality. In this article, we will focus on solving the inequality $5x + 2 \leq 17$.

Understanding the Inequality

The given inequality is $5x + 2 \leq 17$. This means that the expression $5x + 2$ is less than or equal to $17$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign.

Step 1: Subtract 2 from Both Sides

To isolate the term with the variable $x$, we need to subtract 2 from both sides of the inequality. This will give us:

$$5x + 2 - 2 \leq 17 - 2$$

Simplifying the inequality, we get:

$$5x \leq 15$$

Step 2: Divide Both Sides by 5

To isolate the variable $x$, we need to divide both sides of the inequality by 5. This will give us:

$$\frac{5x}{5} \leq \frac{15}{5}$$

Simplifying the inequality, we get:

$$x \leq 3$$

Conclusion

Therefore, the solution to the inequality $5x + 2 \leq 17$ is $x \leq 3$. This means that any value of $x$ that is less than or equal to 3 will satisfy the inequality.

Graphical Representation

The solution to the inequality $5x + 2 \leq 17$ can be represented graphically on a number line. The number line will have a closed circle at $x = 3$ to indicate that $x = 3$ is included in the solution set. The number line will also have an arrow pointing to the left of $x = 3$ to indicate that all values of $x$ less than 3 are also included in the solution set.

Real-World Applications

Solving inequalities has many real-world applications. For example, in finance, inequalities can be used to determine the maximum amount of money that can be borrowed based on a person's income. In engineering, inequalities can be used to determine the maximum stress that a material can withstand. In medicine, inequalities can be used to determine the maximum dose of a medication that can be administered to a patient.

Tips and Tricks

Here are some tips and tricks for solving inequalities:

• Always isolate the variable on one side of the inequality sign.
• Use inverse operations to isolate the variable.
• Check your solution by plugging it back into the original inequality.
• Use graphical representations to visualize the solution set.

Common Mistakes

Here are some common mistakes to avoid when solving inequalities:

• Not isolating the variable on one side of the inequality sign.
• Not using inverse operations to isolate the variable.
• Not checking the solution by plugging it back into the original inequality.
• Not using graphical representations to visualize the solution set.

Conclusion

Solving inequalities is an important skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can solve inequalities with ease. Remember to always isolate the variable on one side of the inequality sign, use inverse operations to isolate the variable, and check your solution by plugging it back into the original inequality. With practice and patience, you will become proficient in solving inequalities and be able to apply this skill to real-world problems.