Questions: Solve the linear inequality. Express the solution using set-builder notation and interval notation. Graph the solution set. 5x + 4 < 34 What is the solution in set-builder notation? x

Solution Steps

To solve the linear inequality \(5x + 4 < 34\), we need to isolate the variable \(x\). This involves subtracting 4 from both sides and then dividing by 5. Once we have the solution, we can express it in set-builder notation and interval notation.

We start with the given inequality: \[ 5x + 4 < 34 \]

To isolate \( x \), we first subtract 4 from both sides of the inequality: \[ 5x + 4 - 4 < 34 - 4 \] \[ 5x < 30 \]

Next, we divide both sides of the inequality by 5 to solve for \( x \): \[ \frac{5x}{5} < \frac{30}{5} \] \[ x < 6 \]

The solution in set-builder notation is: \[ \{x \mid x < 6\} \]

The solution in interval notation is: \[ (-\infty, 6) \]

To graph the solution set, we draw a number line and shade all the numbers less than 6. We use an open circle at 6 to indicate that 6 is not included in the solution set.

\[ \begin{array}{c} \text{Number Line:} \\ \begin{array}{cccccccccccccccc} \cdots & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \cdots \\ \end{array} \\ \begin{array}{cccccccccccccccc} \cdots & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \bullet & \bullet & \bullet & \bullet & \cdots \\ \end{array} \end{array} \]

Final Answer