Which Inequality Is True For X 20

Which Inequality is True for x ≥ 20? A Comprehensive Exploration

Determining which inequalities hold true for x ≥ 20 requires a systematic approach involving understanding inequality properties and applying them to various scenarios. This article will delve into different types of inequalities, exploring how they relate to the condition x ≥ 20 and providing detailed examples to solidify your understanding. We'll cover linear inequalities, quadratic inequalities, and even introduce the concept of absolute value inequalities within this context.

Understanding Inequalities and the Condition x ≥ 20

Before diving into specific examples, let's establish a foundational understanding. The statement "x ≥ 20" means "x is greater than or equal to 20." This sets a lower bound for the variable x; x can be 20, or any value larger than 20. This condition will be our guiding principle as we evaluate different inequalities.

Key Properties of Inequalities:

Remember these fundamental properties when working with inequalities:

• Addition/Subtraction Property: Adding or subtracting the same number from both sides of an inequality does not change the inequality's direction.
• Multiplication/Division Property: Multiplying or dividing both sides by a positive number does not change the inequality's direction. However, multiplying or dividing by a negative number reverses the inequality's direction (e.g., > becomes <).

Linear Inequalities and x ≥ 20

Let's start with linear inequalities, which are inequalities involving only the first power of the variable x.

Example 1: Is the inequality x + 10 > 30 true for x ≥ 20?

To determine this, we solve the inequality:

x + 10 > 30 x > 30 - 10 x > 20

Since x must be greater than 20, and our condition is x ≥ 20, this inequality is mostly true. It's not true for x=20, but it is true for all values of x greater than 20.

Example 2: Is the inequality 2x - 5 ≥ 35 true for x ≥ 20?

Let's solve the inequality:

2x - 5 ≥ 35 2x ≥ 40 x ≥ 20

This inequality is true for all values of x satisfying the condition x ≥ 20.

Example 3: Is the inequality -x + 50 < 30 true for x ≥ 20?

Solving the inequality:

-x + 50 < 30 -x < -20 x > 20

This inequality is true for all values of x greater than 20, but not for x = 20. Therefore it is not entirely true for our condition x ≥ 20

Quadratic Inequalities and x ≥ 20

Quadratic inequalities involve the second power of the variable x (x²). Solving these requires a slightly different approach.

Example 4: Is the inequality x² - 400 ≥ 0 true for x ≥ 20?

This inequality can be factored as:

(x - 20)(x + 20) ≥ 0

This inequality is true when both factors are positive or both are negative.

• Case 1: Both factors positive: x - 20 ≥ 0 and x + 20 ≥ 0. This implies x ≥ 20 and x ≥ -20. Combining these, we get x ≥ 20.
• Case 2: Both factors negative: x - 20 ≤ 0 and x + 20 ≤ 0. This implies x ≤ 20 and x ≤ -20. Combining these, we get x ≤ -20.

Therefore, the inequality x² - 400 ≥ 0 is true when x ≥ 20 or x ≤ -20. Considering our condition x ≥ 20, this inequality is true.

Example 5: Is the inequality -x² + 400 > 0 true for x ≥ 20?

This inequality can be rewritten as:

x² < 400

Taking the square root of both sides:

-20 < x < 20

Given that our condition is x ≥ 20, this inequality is false.

Absolute Value Inequalities and x ≥ 20

Absolute value inequalities introduce another layer of complexity.

Example 6: Is the inequality |x - 20| ≤ 10 true for x ≥ 20?

The inequality |x - 20| ≤ 10 means that the distance between x and 20 is less than or equal to 10. This translates to:

-10 ≤ x - 20 ≤ 10

Adding 20 to all parts:

10 ≤ x ≤ 30

Since this range is contained within x ≥ 20, the inequality is true for all x values satisfying x ≥ 20 that also meet the condition 10 ≤ x ≤ 30.

Example 7: Is the inequality |x - 30| > 10 true for x ≥ 20?

This inequality means the distance between x and 30 is greater than 10. This translates to:

x - 30 > 10 or x - 30 < -10

Solving these:

x > 40 or x < 20

Given our condition x ≥ 20, only the part x > 40 is relevant. Therefore, the inequality is not entirely true for all x ≥ 20; only the values greater than 40 satisfy this condition.

Conclusion: A Systematic Approach to Inequality Evaluation

Determining whether an inequality is true for x ≥ 20 requires a methodical approach:

1. Solve the inequality: Isolate the variable x to determine the range of values that satisfy the inequality.
2. Compare the solution to the condition x ≥ 20: Does the solution set completely overlap with or fully encompass the condition x ≥ 20? If so, the inequality is true for x ≥ 20. If only a portion of the solution set aligns with x ≥ 20, or if there's no overlap, the inequality is not entirely true for all x ≥ 20.
3. Consider all cases: For quadratic and absolute value inequalities, you might need to consider different cases (positive/negative factors, different solution ranges) to establish whether the condition is met.

By following this approach, you can confidently analyze various inequalities and determine their truth value within the specified constraint x ≥ 20. Remember to carefully examine the solution set and compare it to the given condition to achieve accurate results. This systematic approach ensures a thorough and accurate analysis of inequalities within the specified parameter. Understanding these concepts provides a strong foundation for more advanced mathematical problems.