Problem 28 - Precalculus Mathematics for Calculus

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$5 \leq 3 x-4 \leq 14$$

Step-by-Step Solution

Answer

The solution is \([3, 6]\).

Step 1: Understand the Inequality Structure

The inequality given is a compound double inequality: \( 5 \leq 3x - 4 \leq 14 \). This means that \( 3x - 4 \) must be greater than or equal to 5 and less than or equal to 14.

Step 2: Solve the First Inequality

The first part of the inequality is \( 5 \leq 3x - 4 \). Start by adding 4 to both sides to isolate the term with \( x \): \( 5 + 4 \leq 3x \), which simplifies to \( 9 \leq 3x \). Then, divide both sides by 3: \( \frac{9}{3} \leq x \), which simplifies to \( 3 \leq x \).

Step 3: Solve the Second Inequality

The second part of the inequality is \( 3x - 4 \leq 14 \). Add 4 to both sides to isolate the term with \( x \): \( 3x - 4 + 4 \leq 14 + 4 \), which simplifies to \( 3x \leq 18 \). Then, divide both sides by 3: \( x \leq \frac{18}{3} \), which simplifies to \( x \leq 6 \).

Step 4: Combine the Solutions

Now combine the results from both parts. We have \( 3 \leq x \leq 6 \). This means \( x \) can be any number between 3 and 6, inclusive of both 3 and 6.

Step 5: Express the Solution in Interval Notation

Using interval notation, the solution \( 3 \leq x \leq 6 \) is written as \([3, 6]\).

Step 6: Graph the Solution Set

To graph the solution on a number line, draw a solid line between 3 and 6. Place closed circles (or solid dots) on both 3 and 6 to show that they are included in the solution set. This represents all values \( x \) such that \( x \geq 3 \) and \( x \leq 6 \).

Key Concepts

Interval Notation

Interval notation is a concise way to express ranges of values for a variable.
It uses brackets and parentheses to show whether endpoints are included or excluded.
In our solution, we determined that the value of \( x \) lies between 3 and 6, inclusive.
Here's how we express that using interval notation:

• Use square brackets: Square brackets \([\, ]\) are used when the endpoint is included. For example, \([3, 6]\) means 3 and 6 are part of the solution set.
• Use parentheses: Parentheses \((\, )\) are used when the endpoint is not included, such as in \((3, 6)\), which would mean the values can approach 3 and 6 but do not include them.
• In our problem, the solution is \([3, 6]\) indicating that \( x \) can be any number between 3 and 6, including both 3 and 6.

Compound Inequality

A compound inequality involves two separate inequalities joined by the word "and" or "or".
Compound inequalities can look intimidating at first, but they break down into simpler parts.
Let's explore this concept using our exercise:

• In our problem, \( 5 \leq 3x - 4 \leq 14 \) is a compound inequality connected by "and" because it means both conditions must be satisfied simultaneously.
• To solve, we separate it into two parts: \( 5 \leq 3x - 4 \) and \( 3x - 4 \leq 14 \).
• Solving these independently gives us two conditions: \( x \geq 3 \) and \( x \leq 6 \).
• The solution to the compound inequality is the intersection of these two conditions: \( 3 \leq x \leq 6 \).

This means \( x \) must satisfy both inequalities to be true within the same range.

Inequality Graphing

Graphing inequalities can help visually understand the solution set for a problem.
This process involves drawing number lines and shading areas to denote valid solutions.
Here's how we graph a compound inequality:

• For the solution \( 3 \leq x \leq 6 \), you would start by drawing a number line that includes the numbers 3 and 6.
• Closed circles: Place closed or solid circles on the numbers 3 and 6 to indicate they are included in the solution. This corresponds to the square brackets in interval notation.
• Shaded line: Shade the line between these two points (from 3 to 6) to show that \( x \) can be any value within this interval.
• This method effectively shows that the interval includes all numbers from 3 to 6, offering a clear picture of all possible solutions.

Graphing is especially useful for complex inequalities as it provides a quick view of the range of possible values.