Finding the Minimum Value of $f(x) = |x| + |2x + 3|$

To find the minimum value of the function $f(x) = |x| + |2x + 3|$, we analyze the function in different intervals based on the points where the expressions inside the absolute value signs change sign.

The critical points are found by setting the expressions equal to zero:

• $x = 0$
• $2x + 3 = 0 \implies 2x = -3 \implies x = -3/2$

These points divide the number line into three intervals: $x < -3/2$, $-3/2 \le x < 0$, and $x \ge 0$. We define the function $f(x)$ in each interval:

Interval 1: $x < -3/2$

• In this interval, $x$ is negative, so $|x| = -x$.
• $2x + 3$ is negative (since $x < -3/2 \implies 2x < -3 \implies 2x + 3 < 0$), so $|2x + 3| = -(2x + 3)$.
• Therefore, $f(x) = (-x) - (2x + 3) = -x - 2x - 3 = -3x - 3$.
• This function is linear and decreasing. As $x$ approaches $-3/2$, $f(x)$ approaches $-3(-3/2) - 3 = 9/2 - 3 = 4.5 - 3 = 1.5$.

Interval 2: $-3/2 \le x < 0$

• In this interval, $x$ is negative, so $|x| = -x$.
• $2x + 3$ is non-negative (since $x \ge -3/2 \implies 2x \ge -3 \implies 2x + 3 \ge 0$), so $|2x + 3| = 2x + 3$.
• Therefore, $f(x) = (-x) + (2x + 3) = x + 3$.
• This function is linear and increasing. The minimum value in this interval occurs at the smallest value of $x$, which is $x = -3/2$.
• $f(-3/2) = -3/2 + 3 = -1.5 + 3 = 1.5$.

Interval 3: $x \ge 0$

• In this interval, $x$ is non-negative, so $|x| = x$.
• $2x + 3$ is positive (since $x \ge 0 \implies 2x+3 \ge 3$), so $|2x + 3| = 2x + 3$.
• Therefore, $f(x) = x + (2x + 3) = 3x + 3$.
• This function is linear and increasing. The minimum value in this interval occurs at the smallest value of $x$, which is $x = 0$.
• $f(0) = 3(0) + 3 = 3$.

Determining the Overall Minimum Value

Comparing the behavior in the three intervals:

• The function approaches $1.5$ as $x \to -3/2$ from the left.
• The function reaches a minimum value of $1.5$ at $x = -3/2$.
• The function reaches a minimum value of $3$ at $x = 0$ and increases thereafter.

The overall minimum value of the function $f(x) = |x| + |2x + 3|$ is $1.5$. Rounded to one decimal place, the value is $1.5$.