Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

Are the inequalities \(x-4<5\) and \(x>9\) equivalent?

Step-by-Step Solution

Answer

No, the inequalities \(x-4<5\) and \(x>9\) are not equivalent.

Step 1: Solve the first inequality

To solve the inequality \(x-4<5\), add 4 to both sides. The solution is \(x<9\). The solution set includes all real numbers less than 9.

Step 2: Check the second inequality

The second inequality given is \(x>9\). The solution to this inequality is all real numbers greater than 9.

Step 3: Compare the solution sets

As the solution sets for the two inequalities are completely different (one is less than 9 and the other is greater than 9), it can be concluded that the inequalities \(x-4<5\) and \(x>9\) are not equivalent.

Key Concepts

Solving Inequalities

Solving inequalities involves finding all values of a variable that make the inequality true. Unlike equations, which have specific solutions, inequalities describe ranges of values. Let's consider the inequality \( x - 4 < 5 \).To solve it:

• First, we isolate the variable \( x \) by performing operations that maintain the inequality. Here, we add 4 to both sides.
• This results in \( x < 9 \).

The process involves simple arithmetic operations, such as addition, subtraction, multiplication, or division, while respecting the rules of inequality.
Remember to reverse the inequality sign when you multiply or divide both sides by a negative number. This is crucial to getting the correct solution.

Solution Set

The solution set of an inequality consists of all values that satisfy the inequality. For the inequality \( x < 9 \), the solution set includes:

• All real numbers less than 9.

When we say "all real numbers less than 9," we imagine starting from negative infinity and moving up towards but not reaching 9.
It's like a number line that extends infinitely to the left.
The form of the solution set is often expressed using interval notation: \[ (-\infty, 9) \]. For the inequality \( x > 9 \), the solution set includes all real numbers greater than 9. It is denoted as:

• \((9, +\infty)\), which shows that the actual number 9 is not included, but all numbers greater than 9 are part of the set.

Understanding solution sets helps visualize the span of solutions for inequalities.

Equivalent Inequalities

Equivalent inequalities are different inequalities that have the same solution set. In simpler terms, no matter how you write them, they represent the same range of solutions. However, not all inequalities are equivalent.
For instance, let's look at the inequalities \( x - 4 < 5 \) and \( x > 9 \).

• The first inequality, once solved, gives a solution set of all numbers less than 9.
• The second inequality gives a solution set of all numbers greater than 9.

Since these solution sets do not overlap and describe different ranges, the inequalities are not equivalent. Understanding whether inequalities are equivalent helps ensure that the correct comparisons and conclusions are made when working with different sets of inequalities.