Problem 48 - Algebra for College Students

Question

Solve each compound inequality using the compact form. Express the solution sets in interval notation. \(-25 \leq 4 x+3 \leq 19\)

Step-by-Step Solution

Answer

The solution in interval notation is \([-7, 4]\).

Step 1: Break Down the Compound Inequality

The inequality \(-25 \leq 4x + 3 \leq 19\) means we have two separate inequalities: \(-25 \leq 4x + 3\) and \(4x + 3 \leq 19\). We'll solve each inequality separately.

Step 2: Solve the First Inequality

We start with the inequality \(-25 \leq 4x + 3\). First, subtract 3 from both sides to isolate the term containing \(x\): \(-25 - 3 \leq 4x\), which simplifies to \(-28 \leq 4x\). Next, divide both sides by 4 to solve for \(x\): \(-7 \leq x\).

Step 3: Solve the Second Inequality

Now, consider the inequality \(4x + 3 \leq 19\). Subtract 3 from both sides: \(4x \leq 16\). Then, divide by 4 to solve for \(x\): \(x \leq 4\).

Step 4: Combine the Solutions

We now have two conditions for \(x\): \(-7 \leq x\) and \(x \leq 4\). Combine these to find the solution set: \(-7 \leq x \leq 4\).

Step 5: Express in Interval Notation

The solution \(-7 \leq x \leq 4\) in interval notation is \([-7, 4]\). This means \(x\) can be any number from \(-7\) to \(4\), including \(-7\) and \(4\).

Key Concepts

Interval Notation

Understanding interval notation is essential when dealing with inequalities. It provides a concise way to describe a set of numbers that fall within a certain range. This system is particularly useful in mathematics and can simplify communication about ranges of values.
Interval notation uses brackets and parentheses to show which numbers are included in the set:

• Square brackets, like \([a, b]\), indicate that both endpoints \(a\) and \(b\) are included in the set.
• Parentheses, like \(a, b\), indicate that the endpoints are not included.

For example, the interval \([-7, 4]\) means that all numbers from \(-7\) to \(4\) are included, including \(-7\) and \(4\). This is because square brackets are used on both sides.
It's important to remember these differences, as they convey specific information about the solution sets derived from inequalities.

Solving Inequalities

Solving inequalities is a crucial skill in algebra. They help express a relationship where one quantity is either greater than or less than another. Rather than just finding one number, solving inequalities means finding a whole range of numbers that satisfy the given conditions.
There are important rules when solving inequalities:

• When adding or subtracting the same number from both sides, the inequality sign remains unchanged.
• When multiplying or dividing by a positive number, the inequality sign stays the same.
• When multiplying or dividing by a negative number, the inequality sign reverses direction.

Let's consider the inequality \(-25 \leq 4x + 3 \leq 19\). This needs to be split into two separate inequalities. First is \(-25 \leq 4x + 3\), and the second is \(4x + 3 \leq 19\). Solving these gives the possible values of \(x\), ensuring the solution respects all parts of the compound inequality.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations. They form the building blocks for more complex equations and inequalities. In this exercise, the expression \(4x + 3\) is at the center of the inequality that needed solving.
The process involves understanding the expression's components:

• \(x\) represents the variable we want to solve for.
• \(4x\) is a term showing \(x\) being multiplied by \(4\).
• \(+ 3\) is a constant being added to \(4x\).

When working with algebraic expressions in the context of inequalities, the goal is to isolate the variable on one side of the inequality. This usually requires inverse operations, such as subtracting and dividing, to simplify the expression step by step.
By breaking down the expression \(4x + 3\) into its parts during the solving process, you gain a better understanding of how each part influences the inequality and contributes to finding the range of possible solutions.