1. **Problem Statement:** (a) Sketch the graph of $y = |2x + 1|$. (b) Solve the inequality $3x + 5 < |2x + 1|$. 2. **Understanding the absolute value function:** The function $y = |2x + 1|$ is an absolute value function, which creates a "V" shape graph. The vertex occurs where the inside of the absolute value is zero: $$2x + 1 = 0 \implies x = -\frac{1}{2}$$ 3. **Piecewise definition of the function:** For $x \geq -\frac{1}{2}$, the function is: $$y = 2x + 1$$ For $x < -\frac{1}{2}$, the function is: $$y = -(2x + 1) = -2x - 1$$ 4. **Sketching the graph:** - Plot the vertex at $\left(-\frac{1}{2}, 0\right)$. - For $x \geq -\frac{1}{2}$, graph the line $y = 2x + 1$. - For $x < -\frac{1}{2}$, graph the line $y = -2x - 1$. 5. **Solving the inequality $3x + 5 < |2x + 1|$:** We consider two cases based on the definition of the absolute value. **Case 1: $x \geq -\frac{1}{2}$** $$3x + 5 < 2x + 1$$ Subtract $2x + 1$ from both sides: $$3x + 5 - (2x + 1) < 0$$ $$x + 4 < 0$$ $$x < -4$$ But this contradicts $x \geq -\frac{1}{2}$, so no solution in this case. **Case 2: $x < -\frac{1}{2}$** $$3x + 5 < -2x - 1$$ Add $2x + 1$ to both sides: $$3x + 5 + 2x + 1 < 0$$ $$5x + 6 < 0$$ $$5x < -6$$ $$x < -\frac{6}{5}$$ Since $x < -\frac{1}{2}$, the solution here is: $$x < -\frac{6}{5}$$ 6. **Final solution:** The solution to the inequality is: $$\boxed{x < -\frac{6}{5}}$$