Problem 31 - College Algebra

Question

For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation. $$ 2 x-5<-11 \text { or } 5 x+1 \geq 6 $$

Step-by-Step Solution

Answer

\( (-\infty, -3) \cup [1, \infty) \)

Step 1: Solve the First Inequality

The first inequality is \(2x - 5 < -11\). To solve for \(x\), start by adding 5 to both sides to isolate the term with \(x\): \(2x < -11 + 5\) This simplifies to: \(2x < -6\). Now, divide both sides by 2: \(x < -3\).

Step 2: Solve the Second Inequality

The second inequality is \(5x + 1 \geq 6\). Start by subtracting 1 from both sides to get: \(5x \geq 6 - 1\) This simplifies to: \(5x \geq 5\). Next, divide both sides by 5: \(x \geq 1\).

Step 3: Combine the Solutions

The solutions from Step 1 and Step 2 are \(x < -3\) and \(x \geq 1\), respectively. Since the original problem is a compound inequality with the word 'or,' we take the union of these solutions. This means we include values of \(x\) that satisfy either inequality.

Step 4: Express the Solution in Interval Notation

Since the solution is \(x < -3 \text{ or } x \geq 1\), we write it in interval notation as: \( (-\infty, -3) \cup [1, \infty) \). This interval notation indicates all numbers less than -3, and all numbers greater than or equal to 1.

Key Concepts

Inequality Signs

Inequality signs are fundamental symbols used to compare values, and they can often be seen in mathematical expressions that determine the range of possible solutions. There are four primary inequality signs:

• "<" means "less than."
• "≤" stands for "less than or equal to."
• ">" signifies "greater than."
• "≥" means "greater than or equal to."

These inequalities define ranges rather than exact values.
For example, the inequality \(2x - 5 < -11\) means that the expression \(2x - 5\) must result in a value that is less than -11. Similarly, the inequality \(5x + 1 \geq 6\) suggests that the expression \(5x + 1\) must be at least 6.
Understanding these signs is crucial for grasping how to manipulate and solve inequalities.

Interval Notation

Interval notation is a way of writing subsets of the real numbers and is particularly helpful in expressing solutions to inequalities. It provides a compact and precise way to describe the set of all numbers between given endpoints, which can be represented with brackets.

• "(" and ")" indicate that the endpoints are not included (open interval).
• "[" and "]" indicate that the endpoints are included (closed interval).

For the compound inequality given in the problem, the solution was \(x < -3\) or \(x \geq 1\). Converted to interval notation, this becomes \((-\infty, -3) \cup [1, \infty)\).
The use of \((-\infty\) and \(\infty)\) signifies an unbounded direction, meaning the interval continues indefinitely. So for \(x < -3\), "non-inclusive" parentheses denote that -3 itself is not included, while for \(x \geq 1\), the square bracket "[" denotes that 1 is part of the solution set. Understanding interval notation helps you to clearly express the solutions of inequalities in a symbolic form.

Solving Inequalities

Solving inequalities is like solving equations, but the main difference is that you need to be mindful of the inequality signs throughout the process. Let's break this down using the steps in our exercise.

In the first inequality, \(2x - 5 < -11\), begin by isolating the variable \(x\). Here, you would add 5 to both sides, leading to \(2x < -6\). The next step is to divide both sides by 2, which simplifies it to \(x < -3\).

Similarly, for the second inequality, \(5x + 1 \geq 6\), start by subtracting 1 from both sides, giving \(5x \geq 5\). Divide both sides by 5, resulting in \(x \geq 1\).

When dealing with compound inequalities, it is essential to recognize whether the connection between them is "or" or "and." The problems here involve "or," indicating that any value that satisfies at least one of the inequalities is a solution. Thus, the solution combines the results: \(x < -3\) or \(x \geq 1\).
Solving inequalities requires careful consideration of operations on both sides, particularly when multiplying or dividing by negative numbers, as these reverse the inequality signs. Mastering these techniques allows you to tackle various inequality problems effectively.