Solve The Inequality:${ -2 \leq 3 - X \leq 5 }$

Mar 12, 2025 by ADMIN 49 views

Solving the Inequality: A Step-by-Step Guide

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Introduction

In this article, we will delve into the world of inequalities and explore a specific problem: solving the inequality $-2 \leq 3 - x \leq 5$ . Inequalities are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various fields, including algebra, calculus, and statistics. In this discussion, we will break down the solution to this inequality, providing a clear and concise explanation of each step.

Understanding the Inequality

The given inequality is a compound inequality, which means it consists of two separate inequalities joined by the word "and." In this case, we have:

-2 \leq 3 - x \leq 5

To solve this inequality, we need to isolate the variable $x$ and find the range of values that satisfy the inequality.

Step 1: Isolate the Variable

The first step in solving the inequality is to isolate the variable $x$ . We can do this by subtracting 3 from all three parts of the inequality:

-2 - 3 \leq 3 - x - 3 \leq 5 - 3

This simplifies to:

-5 \leq -x \leq 2

Step 2: Multiply by -1

To isolate the variable $x$ , we need to multiply both sides of the inequality by -1. However, when we multiply an inequality by a negative number, we need to reverse the direction of the inequality signs:

-(-5) \geq -(-x) \geq -2

This simplifies to:

5 \geq x \geq -2

Step 3: Write the Solution in Interval Notation

The solution to the inequality can be written in interval notation as:

[-2, 5]

This means that the value of $x$ can be any real number between -2 and 5, inclusive.

Conclusion

Solving the inequality $-2 \leq 3 - x \leq 5$ requires careful manipulation of the inequality signs and the variable $x$ . By following the steps outlined above, we can isolate the variable and find the range of values that satisfy the inequality. The solution to this inequality is $[-2, 5]$ , which means that the value of $x$ can be any real number between -2 and 5, inclusive.

Frequently Asked Questions

Q: What is the difference between a compound inequality and a single inequality?

A: A compound inequality consists of two or more separate inequalities joined by the word "and" or "or." A single inequality, on the other hand, consists of a single inequality statement.

Q: How do I know which direction to reverse the inequality signs when multiplying by a negative number?

A: When multiplying an inequality by a negative number, you need to reverse the direction of the inequality signs. For example, if you have the inequality $a \leq b$ , and you multiply both sides by -1, the resulting inequality would be $-b \leq -a$ .

Q: Can I use interval notation to write the solution to any inequality?

A: Yes, interval notation can be used to write the solution to any inequality. However, the interval notation may vary depending on the type of inequality and the solution.

Additional Resources

For more information on solving inequalities, including compound inequalities, check out the following resources:

Khan Academy: Solving Inequalities
Mathway: Solving Inequalities
Wolfram Alpha: Solving Inequalities

Final Thoughts

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Introduction

In our previous article, we explored the solution to the inequality $-2 \leq 3 - x \leq 5$ . In this article, we will delve into a Q&A format, providing answers to common questions related to solving inequalities. Whether you're a student, teacher, or simply looking to brush up on your math skills, this article is for you.

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. 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.

Q: How do I solve a linear inequality with two variables?

A: To solve a linear inequality with two variables, you need to isolate one of the variables. For example, if you have the inequality $2x + 3y \leq 5$ , you can isolate $x$ by subtracting $3y$ from both sides:

2x \leq 5 - 3y

Then, you can divide both sides by 2 to get:

x \leq \frac{5 - 3y}{2}

Q: What is the difference between a compound inequality and a double inequality?

A: A compound inequality is an inequality that consists of two or more separate inequalities joined by the word "and" or "or." A double inequality, on the other hand, is an inequality that consists of two separate inequalities joined by a single inequality sign, such as $\leq$ or $\geq$ .

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign chart method to determine the intervals where the inequality is true. For example, if you have the inequality $x^2 + 4x + 4 \leq 0$ , you can factor the quadratic expression as:

(x + 2)^2 \leq 0

Then, you can use the sign chart method to determine that the inequality is true only when $x = -2$ .

Q: Can I use the quadratic formula to solve a quadratic inequality?

A: No, the quadratic formula is used to solve quadratic equations, not quadratic inequalities. However, you can use the quadratic formula to find the roots of the quadratic equation, and then use the sign chart method to determine the intervals where the inequality is true.

Q: How do I graph a linear inequality on a number line?

A: To graph a linear inequality on a number line, you need to plot a point on the number line that satisfies the inequality, and then draw a line segment that extends in the direction of the inequality. For example, if you have the inequality $x \geq 2$ , you can plot a point at $x = 2$ and then draw a line segment that extends to the right.

Q: Can I use interval notation to write the solution to a quadratic inequality?

A: Yes, interval notation can be used to write the solution to a quadratic inequality. However, the interval notation may vary depending on the type of inequality and the solution.

Q: How do I determine the intervals where a quadratic inequality is true?

A: To determine the intervals where a quadratic inequality is true, you need to use the sign chart method. This involves plotting the roots of the quadratic equation on a number line, and then using the sign chart to determine the intervals where the inequality is true.

Q: Can I use the quadratic formula to find the roots of a quadratic equation?

A: Yes, the quadratic formula can be used to find the roots of a quadratic equation. The quadratic formula is:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the sign chart method to determine the intervals where a quadratic inequality is true?

A: To use the sign chart method, you need to plot the roots of the quadratic equation on a number line, and then use the sign chart to determine the intervals where the inequality is true. The sign chart is a table that shows the sign of the quadratic expression in each interval.

Q: Can I use the quadratic formula to solve a system of quadratic equations?

A: No, the quadratic formula is used to solve quadratic equations, not systems of quadratic equations. However, you can use the quadratic formula to find the roots of each equation, and then use substitution or elimination to solve the system.

Q: How do I graph a quadratic inequality on a coordinate plane?

A: To graph a quadratic inequality on a coordinate plane, you need to plot the roots of the quadratic equation on the x-axis, and then draw a line segment that extends in the direction of the inequality. You can also use the sign chart method to determine the intervals where the inequality is true.

Q: Can I use the quadratic formula to find the vertex of a quadratic function?

A: Yes, the quadratic formula can be used to find the vertex of a quadratic function. The vertex is the point on the graph where the quadratic function changes direction.

Q: How do I use the quadratic formula to find the axis of symmetry of a quadratic function?

A: To use the quadratic formula to find the axis of symmetry, you need to find the vertex of the quadratic function, and then use the formula:

x = \frac{-b}{2a}

to find the x-coordinate of the axis of symmetry.

Q: Can I use the quadratic formula to solve a system of linear and quadratic equations?

A: No, the quadratic formula is used to solve quadratic equations, not systems of linear and quadratic equations. However, you can use the quadratic formula to find the roots of the quadratic equation, and then use substitution or elimination to solve the system.

Q: How do I graph a system of linear and quadratic equations on a coordinate plane?

A: To graph a system of linear and quadratic equations on a coordinate plane, you need to plot the roots of each equation on the x-axis, and then draw a line segment that extends in the direction of the equation. You can also use the sign chart method to determine the intervals where the inequality is true.

Q: Can I use the quadratic formula to find the intersection points of two quadratic functions?

A: Yes, the quadratic formula can be used to find the intersection points of two quadratic functions. You can use the quadratic formula to find the roots of each equation, and then use substitution or elimination to find the intersection points.

Q: How do I use the quadratic formula to find the maximum or minimum value of a quadratic function?

A: To use the quadratic formula to find the maximum or minimum value of a quadratic function, you need to find the vertex of the quadratic function, and then use the formula:

y = a(x - h)^2 + k

to find the maximum or minimum value.

Q: Can I use the quadratic formula to solve a system of quadratic inequalities?

A: No, the quadratic formula is used to solve quadratic equations, not systems of quadratic inequalities. However, you can use the quadratic formula to find the roots of each equation, and then use substitution or elimination to solve the system.

Q: How do I graph a system of quadratic inequalities on a coordinate plane?

A: To graph a system of quadratic inequalities on a coordinate plane, you need to plot the roots of each equation on the x-axis, and then draw a line segment that extends in the direction of the inequality. You can also use the sign chart method to determine the intervals where the inequality is true.

Q: Can I use the quadratic formula to find the solution to a quadratic equation with complex roots?

A: Yes, the quadratic formula can be used to find the solution to a quadratic equation with complex roots. You can use the quadratic formula to find the roots of the equation, and then use the formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

to find the complex roots.

Q: How do I use the quadratic formula to find the solution to a quadratic equation with irrational roots?

A: To use the quadratic formula to find the solution to a quadratic equation with irrational roots, you need to find the roots of the equation, and then use the formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

to find the irrational roots.

Q: Can I use the quadratic formula to solve a system of quadratic equations with complex roots?

A: Yes, the quadratic formula can be used to solve a system of quadratic equations with complex roots. You can use the quadratic formula to find the roots of each equation, and then use substitution or elimination to solve the system.

Q: How do I graph a system of quadratic equations with complex roots on a coordinate plane?

A: To graph a system of quadratic equations with complex roots on a coordinate plane, you need to plot the roots of each equation on the x-axis, and then draw a line segment that extends in the direction of the equation. You can also use the sign chart method to determine the intervals where the inequality is true.

Q: Can I use the quadratic formula to find the solution to a quadratic equation with rational roots?

A: Yes, the quadratic formula can be used to find the solution to a quadratic equation with rational roots.