Problem 10 - Elementary Algebra

Question

Determine whether the given number is a solution to the given inequality. $$ 4 x-1<-17 \text { or } 3 x+2 \geq 6 ; x=1 $$

Step-by-Step Solution

Answer

x=1 is not a solution to the given inequalities.

Step 1: Plug x into the First Inequality

Start with the inequality \( 4x-1<-17 \). Plug in \( x = 1 \) to get \( 4(1) - 1 < -17 \). This simplifies to \( 4 - 1 = 3 \). Hence, \( 3 < -17 \) which is false.

Step 2: Plug x into the Second Inequality

Now consider the second inequality \( 3x+2 \geq 6 \). Plug in \( x = 1 \) to get \( 3(1) + 2 \geq 6 \). This simplifies to \( 3 + 2 = 5 \). Hence, \( 5 \geq 6 \) which is also false.

Step 3: Conclusion

Since both inequalities are false for \( x = 1 \), it means that \( x = 1 \) is not a solution for either part of the compound inequality.

Key Concepts

Understanding Compound Inequalities

Compound inequalities are expressions that combine two inequalities linked by the words "and" or "or." When using "or," like in our exercise, a solution can satisfy either one or both inequalities. In the inequality given—\(4x-1<-17 \) or \(3x+2 \geq 6\)—this means we are interested in any value of \(x\) that makes at least one of these inequalities true.

Using an "or" compound inequality can be visualized as two separate conditions that can hold independently. This gives more flexibility: as long as one part is true, the whole statement holds true. It's helpful to:

• Check each inequality separately.
• Understand that only one needs to be satisfied for the whole compound inequality to be considered true.

In this exercise, if some \(x\) value can solve either \( 4x-1<-17 \) or \( 3x+2 \geq 6 \), then it satisfies the compound inequality. However, if none satisfy, like in our exercise, then \(x\) does not solve the inequality.

Solution Verification of Inequalities

Solution verification involves plugging a proposed value into the inequality to see if it holds true. It's a straightforward way to check if a solution is correct or not. In our exercise, the value \( x = 1 \) was tested against both parts of the compound inequality.

• For each inequality, substitute the value into the expression.
• Simplify the equation to check if the inequality is satisfied.

Let's break down the steps used:

1. **First Inequality:** \( 4x-1<-17 \). By substituting \( x=1 \), we obtained \( 4 - 1 = 3\). Since \(3 < -17\) is false, this means 1 is not a solution for the first inequality.
2. **Second Inequality:** \( 3x+2 \geq 6\). Substituting \(x=1\) gives \(3 + 2 = 5\), and since \(5 \geq 6\) is false, 1 is not a solution for the second inequality.

This clear check confirms whether a number satisfies a compound inequality by directly verifying each part.

Inequality Simplification Techniques

Simplifying inequalities is a key step in solving them efficiently. This involves combining like terms and transforming the expression into a clearer form. Here’s how it applies to compound inequalities:

**Simplification Steps:**

• Evaluate each side of the inequality separately.
• Perform basic arithmetic operations in the correct order.
• Make the inequality more basic so that it’s easier to judge its truth.

For example, in the inequality \( 4x-1<-17 \), starting by multiplying and then simplifying it helps in assessing the relationship between terms like 3 not being less than -17. Similarly, in \( 3x + 2 \geq 6 \), adding and correcting leads to a straightforward comparison like 5 not being greater than or equal to 6.

Using these steps can make evaluating inequalities more intuitive. Proper simplification is essential to avoid errors and to accurately assess the validity of possible solutions. This clearer representation is crucial, especially when confirming solutions.