Problem 6 - Intermediate Algebra

Question

The double inequality \(4<3 x+5 \leq 15\) is equivalent to \(4<3 x+5,3 x+5 \leq 15\)

Step-by-Step Solution

Answer

The solution is \(-\frac{1}{3} < x \leq \frac{10}{3}\).

Step 1: Understand the Original Double Inequality

The given inequality is \(4 < 3x + 5 \leq 15\). This means that \(3x + 5\) is greater than 4 and at the same time, \(3x + 5\) is less than or equal to 15.

Step 2: Split the Double Inequality Into Two

We can split the original double inequality into two separate inequalities: \(4 < 3x + 5\) and \(3x + 5 \leq 15\). We will solve each of these inequalities separately.

Step 3: Solve the First Inequality

Start by isolating \(3x\) in the inequality \(4 < 3x + 5\). Subtract 5 from both sides to get \(4 - 5 < 3x\), which simplifies to \(-1 < 3x\). Next, divide both sides by 3 to solve for \(x\), resulting in \(-\frac{1}{3} < x\).

Step 4: Solve the Second Inequality

Now, solve the inequality \(3x + 5 \leq 15\). Subtract 5 from both sides to get \(3x \leq 10\). Then, divide both sides by 3 to find \(x \leq \frac{10}{3}\).

Step 5: Combine the Solutions

The solution to the original double inequality is the intersection of the solutions from Step 3 and Step 4. Therefore, \(-\frac{1}{3} < x \leq \frac{10}{3}\).

Key Concepts

Double Inequality

A double inequality is like juggling two inequalities at the same time. It shows that a certain expression falls between two values. Consider the double inequality: \(4 < 3x + 5 \leq 15\). This means that the expression \(3x + 5\) is squeezed between 4 and 15.
This type of inequality statement helps us understand the range within which our variable operates.
To handle a double inequality, we split it into two separate inequalities. For our example, we want to analyze these two:

• \(4 < 3x + 5\)
• \(3x + 5 \leq 15\)

By solving each part separately, we can ensure we don't miss any crucial steps. We'll be able to determine how each inequality impacts our variable \(x\). This process eventually leads us to find a combined solution where both conditions are true at the same time.

Step-by-Step Solutions

Step-by-step solutions are your best friend when tackling math problems. They break down complex processes into simple actions, guiding you towards the solution without missing any details.

Let's apply it to our double inequality problem. We began by understanding the meaning of \(4 < 3x + 5 \leq 15\). Next, we split this into the two inequalities mentioned earlier. From here, we can move on to solve each one:

• First, for \(4 < 3x + 5\): subtract 5 from both sides to get \(3x > -1\) which simplifies to \(x > -\frac{1}{3}\) after dividing by 3.
• Second, for \(3x + 5 \leq 15\): likewise, we subtract 5 from both sides resulting in \(3x \leq 10\), then solve to find \(x \leq \frac{10}{3}\).

Each small, logical move is carefully executed to not lose sight of the overall goal — finding the values of \(x\) that satisfy both conditions.

Inequality Intersection

The concept of inequality intersection is key when solving double inequalities. It's about finding the overlap, or intersection, of solutions from our two single inequalities.

From our previous work, we know:

• \(x > -\frac{1}{3}\)
• \(x \leq \frac{10}{3}\)

The intersection takes both of these into account to find where they meet on a number line. So, what are we left with?
We want the values that work in both situations. The final solution becomes \(-\frac{1}{3} < x \leq \frac{10}{3}\).

This means \(x\) can be any number greater than \(-\frac{1}{3}\) and up to and including \(\frac{10}{3}\). This overlap is crucial in establishing where your variable satisfies both conditions simultaneously, providing a complete solution to your inequality problem.