State All Integer Values Of $x$ In The Interval $-1 \leq X \leq 6$ That Satisfy The Following Inequality:\$5x + 10 \ \textless \ 10$[/tex\]

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Introduction

In mathematics, inequalities are a fundamental concept used to describe relationships between variables. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving a linear inequality and finding the integer values of x in a given interval that satisfy the inequality.

The Inequality

The given inequality is:

5x+10<105x + 10 < 10

Our goal is to find the integer values of x in the interval $-1 \leq x \leq 6$ that satisfy this inequality.

Step 1: Subtract 10 from Both Sides

To isolate the term involving x, we need to subtract 10 from both sides of the inequality.

5x+10−10<10−105x + 10 - 10 < 10 - 10

This simplifies to:

5x<05x < 0

Step 2: Divide Both Sides by 5

Next, we need to divide both sides of the inequality by 5 to solve for x.

5x5<05\frac{5x}{5} < \frac{0}{5}

This simplifies to:

x<0x < 0

Step 3: Find the Integer Values of x in the Interval

Now that we have the inequality $x < 0$, we need to find the integer values of x in the interval $-1 \leq x \leq 6$ that satisfy this inequality.

The integer values of x in the interval $-1 \leq x \leq 6$ are:

−1,−2,−3,−4,−5,−6-1, -2, -3, -4, -5, -6

However, we need to check which of these values satisfy the inequality $x < 0$.

Checking the Values

Let's check each value to see if it satisfies the inequality $x < 0$.

  • -1 < 0$ (True)

  • -2 < 0$ (True)

  • -3 < 0$ (True)

  • -4 < 0$ (True)

  • -5 < 0$ (True)

  • -6 < 0$ (True)

All the values satisfy the inequality $x < 0$.

Conclusion

In conclusion, the integer values of x in the interval $-1 \leq x \leq 6$ that satisfy the inequality $5x + 10 < 10$ are:

−1,−2,−3,−4,−5,−6-1, -2, -3, -4, -5, -6

These values satisfy the inequality $x < 0$, which is the solution to the given inequality.

Final Answer

The final answer is:

\boxed{-1, -2, -3, -4, -5, -6}$<br/> **Solving Inequalities: A Q&A Guide** ===================================== **Introduction** --------------- In our previous article, we solved a linear inequality and found the integer values of x in a given interval that satisfy the inequality. In this article, we will provide a Q&A guide to help you understand the concept of solving inequalities and how to apply it to different types of inequalities. **Q: What is an inequality?** --------------------------- A: An inequality is a statement that describes a relationship between two expressions, where one expression is greater than, less than, or equal to the other expression. **Q: What are the different types of inequalities?** ---------------------------------------------- A: There are two main types of inequalities: * **Linear inequalities**: These are inequalities that involve a linear expression, such as 2x + 3 < 5. * **Non-linear inequalities**: These are inequalities that involve a non-linear expression, such as x^2 + 2x + 1 > 0. **Q: How do I solve a linear inequality?** ----------------------------------------- A: To solve a linear inequality, follow these steps: 1. **Isolate the variable**: Move all terms involving the variable to one side of the inequality. 2. **Simplify the inequality**: Combine like terms and simplify the inequality. 3. **Solve for the variable**: Solve for the variable by dividing or multiplying both sides of the inequality by a constant. **Q: How do I solve a non-linear inequality?** --------------------------------------------- A: To solve a non-linear inequality, follow these steps: 1. **Factor the expression**: Factor the non-linear expression into a product of two or more factors. 2. **Set each factor equal to zero**: Set each factor equal to zero and solve for the variable. 3. **Determine the sign of the expression**: Determine the sign of the expression in each interval. **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. A **non-strict inequality** is an inequality that uses the symbols ≤ or ≥, such as x ≤ 2. **Q: How do I graph an inequality on a number line?** ------------------------------------------------ A: To graph an inequality on a number line, follow these steps: 1. **Determine the direction of the inequality**: Determine the direction of the inequality by looking at the symbol. 2. **Mark the critical point**: Mark the critical point on the number line. 3. **Shade the correct region**: Shade the region that satisfies the inequality. **Q: What is the relationship between inequalities and equations?** ---------------------------------------------------------------- A: Inequalities and equations are related in that they both describe relationships between variables. However, inequalities describe relationships that are not equal, while equations describe relationships that are equal. **Conclusion** ---------- In conclusion, solving inequalities is an important concept in mathematics that involves finding the values of the variable that satisfy the given inequality. By following the steps outlined in this Q&A guide, you can learn how to solve different types of inequalities and apply it to real-world problems. **Final Answer** -------------- The final answer is: * **Linear inequalities**: Solve by isolating the variable, simplifying the inequality, and solving for the variable. * **Non-linear inequalities**: Solve by factoring the expression, setting each factor equal to zero, and determining the sign of the expression. * **Graphing inequalities**: Graph by determining the direction of the inequality, marking the critical point, and shading the correct region.