1. **State the problem:** You have points A, B, C, and D on a line segment. Lengths are given as: $$AB = x, \quad BC = x + 1, \quad CD = 2x.$$ The length of the entire segment \(AD = 19\) cm. (a) Show that $$4x + 1 = 19$$ (b) Solve the equation $$4x + 1 = 19$$ (c) Find the length of segment \(BD\). --- 2. **Part (a): Show that $$4x + 1 = 19$$** The total length $$AD$$ is the sum of the segments: $$AD = AB + BC + CD = x + (x + 1) + 2x.$$ Simplify this: $$x + x + 1 + 2x = (1x + 1x + 2x) + 1 = 4x + 1.$$ Given that $$AD = 19$$, we thus have: $$4x + 1 = 19.$$ 3. **Part (b): Solve $$4x + 1 = 19$$** Subtract 1 from both sides: $$4x + 1 - 1 = 19 - 1 \Rightarrow 4x = 18.$$ Divide both sides by 4: $$x = \frac{18}{4} = 4.5.$$ 4. **Part (c): Find \(BD\)** Length $$BD = BC + CD = (x + 1) + 2x = x + 1 + 2x = 3x + 1.$$ Substitute $$x = 4.5$$: $$BD = 3(4.5) + 1 = 13.5 + 1 = 14.5.$$ **Final answer:** (a) $4x + 1 = 19$ (b) $x = 4.5$ (c) $BD = 14.5$ cm