Problem 30 - Precalculus Mathematics for Calculus

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$1<3 x+4 \leq 16$$

Step-by-Step Solution

Answer

The solution is \(( -1, 4 ]\) in interval notation.

Step 1: Simplify the Inequality

Start by addressing the compound inequality, which includes two parts: \(1 < 3x + 4\) and \(3x + 4 \leq 16\). We'll handle these separately initially to simplify each part.

Step 2: Solve the First Part

For \(1 < 3x + 4\), subtract 4 from both sides to isolate the term with \(x\): \[1 - 4 < 3x\]\[-3 < 3x\]Next, divide both sides by 3:\[-1 < x\].

Step 3: Solve the Second Part

For \(3x + 4 \leq 16\), subtract 4 from both sides to isolate the \(3x\) term:\[3x \leq 16 - 4\]\[3x \leq 12\]Divide both sides by 3:\[x \leq 4\].

Step 4: Combine the Solutions

Now, combine the two results \(-1 < x\) and \(x \leq 4\). This gives us the compound inequality:\(-1 < x \leq 4\).

Step 5: Express in Interval Notation

Express the solution \(-1 < x \leq 4\) using interval notation:The solution set in interval notation is \(( -1, 4 ]\).

Step 6: Graph the Solution Set

Draw a number line and represent the solution \(( -1, 4 ]\):- Draw an open circle (or hollow dot) at \(-1\), indicating that \(-1\) is not included.- Draw a closed circle (or solid dot) at \(4\), indicating \(4\) is included.- Shade the region in between \(-1\) and \(4\) on the number line.

Key Concepts

interval notation

Interval notation is a way of representing a range of values, often used in expressing solutions to inequalities. In this form, a combination of brackets and parentheses indicates which endpoints are included or excluded.

Here's a quick guideline:

• ")" and "(" are used for numbers that are not part of the solution set, indicating that the endpoint is not included.
• "]" and "[" denote that the number is included in the solution set.

To illustrate, given the inequality \(-1 < x \leq 4\), the interval notation is given as \( (-1, 4] \).

In this notation:

• \((-1, 4]\) signifies that the interval starts just above \(-1\) and includes the number \(4\).
• The parenthesis \("("\) next to \(-1\) indicates that \(-1\) is not part of the solution, matching the "less than" sign in the inequality.
• The bracket \("]"\) next to \(4\) signifies that \(4\) is included, as confirmed by the "less than or equal to" symbol in the original inequality.

This concise representation not only simplifies writing solutions to inequalities, It also makes it easy to visualize the set of possible solutions.

compound inequality

A compound inequality occurs when two separate inequalities are combined into one expression, showing the range of values that a variable can take. In our exercise, we are working with the compound inequality \(1 < 3x + 4 \leq 16\).

This expression actually consists of two distinct inequalities:

• \(1 < 3x + 4\)
• \(3x + 4 \leq 16\)

The trick to solving compound inequalities is to solve each part separately. Then, the solution is given by those values of \(x\) that satisfy both inequalities.

In practice:

• We solve \(1 < 3x + 4\) and reduce it to \(-1 < x\), meaning \(x\) must be greater than \(-1\).
• Next, solve \(3x + 4 \leq 16\) to find \(x \leq 4\),
• Combine these solutions to find a range for \(x\): \(-1 < x \leq 4\).

Compound inequalities require careful attention to the logic of combining solutions. Each part of the compound must be true for the final inequality to hold.

solution set

The solution set of an inequality includes all the values that make the inequality true. Once the expression is solved, like in our example, \(-1 < x \leq 4\), the solution set captures all possible values for \(x\) within the specified range.

The solution set gives us pivotal information:

• It represents all input values (in this case, \(x\)), that satisfy the inequality.
• It creates a bridge between an abstract algebraic expression and tangible numbers or intervals.
• Visualizing it on a number line often helps comprehend the range of possible solutions.

In our exercise, the solution set is not just numbers but a defined segment of the number line, \((-1, 4]\).

Representing this interval graphically involves several steps:

• We draw an open circle at \(-1\), showing it's not included in the solution set.
• We place a closed circle at \(4\), as this endpoint is included.
• Between these points, we shade the region to illustrate all numbers \(x\) that lie within the range.

Thus, a solution set not only determines the answers but presents them in a comprehensible visual form, ensuring clarity on the issue of which values satisfy the given inequality.