Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. 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 blank is filled with "\(3x + 5 \leq 15\)".

Step 1: Understand the Double Inequality

The double inequality given is \(4 < 3x + 5 \leq 15\). This can be broken down into two separate inequalities: \(4 < 3x + 5\) and \(3x + 5 \leq 15\). The problem states that the first inequality is \(4 < 3x + 5\) and asks for the second part.

Step 2: Identify the Remaining Part

After recognizing the first inequality \(4 < 3x + 5\), the second remaining part of the double inequality is the one involving the right side of the original inequality: \(3x + 5 \leq 15\). This completes the double inequality by providing the upper boundary condition.

Step 3: Fill in the Blank

The blank should therefore be filled in with the inequality: \(3x + 5 \leq 15\). So the complete equivalent statement for the double inequality is \(4 < 3x + 5\) and \(3x + 5 \leq 15\).

Key Concepts

Double Inequalities

Double inequalities are expressions that have two inequality symbols. They involve three parts, connecting them with logical boundaries. Consider the inequality given: \(4 < 3x + 5 \leq 15\). This reads as the middle expression \(3x + 5\) being greater than 4 and less than or equal to 15. This translates into two separate but related inequalities:

• The first: \(4 < 3x + 5\)
• The second: \(3x + 5 \leq 15\)

By analyzing both parts, you can understand the range within which \(3x + 5\) must lie. This type of problem requires ensuring that both conditions are satisfied simultaneously. It’s like finding a sweet spot within a defined range. This dual condition ensures that your solution covers the complete valid range of the equation.

Inequality Manipulation

Manipulating inequalities involves adjusting them while maintaining their truth. Similar to equations, you can add, subtract, multiply, or divide both sides. However, the direction of the inequality sign must be carefully watched, especially when multiplying or dividing by a negative number.
When breaking down the double inequality \(4 < 3x + 5 \leq 15\), we handle each inequality individually. Start with the first inequality \(4 < 3x + 5\) and subtract 5 from both sides:

• \(4 - 5 < 3x\)

Which simplifies to \(-1 < 3x\). For the second part, \(3x + 5 \leq 15\), also subtract 5:

• \(3x + 5 - 5 \leq 15 - 5\)

Resulting in \(3x \leq 10\). Now both inequalities have the form \(-1 < 3x\) and \(3x \leq 10\). With consistent manipulation, you can solve for \(x\) by dividing by 3, provided no negative numbers are involved to flip the inequality symbols.

Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols instead of an equal sign. They represent conditions or constraints within values, not just specific solutions. Like equations, they can be graphed, usually as a shaded region of the number line or plane, showing all possible solutions.
In the example \(-1 < 3x \leq 10\), the solution set can be found by isolating \(x\) in both inequalities. Dividing both parts by 3 gives you:

• \(\frac{-1}{3} < x\) for the first inequality
• \(x \leq \frac{10}{3}\) for the second inequality

Together, they indicate that the value of \(x\) lies between \(-\frac{1}{3}\) and \(\frac{10}{3}\). Understanding these principles helps you graphically represent and contextually understand where the solution lives in relation to the real-number line, or determine where decisions are valid within real-world thresholds.