B2 Solve the following inequalities. Represent the solution sets on separate number lines. (a) \( (1+x)(6-x) \leqslant-8 \) (b) \( 2 x(x+2)<(x+1)(x+3) \)

**(a)** We start with the inequality \[ (1+x)(6-x) \leq -8. \] *Step 1: Expand the left-hand side.* \[ (1+x)(6-x)=6+5x-x^2. \] *Step 2: Rewrite the inequality.* \[ 6+5x-x^2 \le -8. \] *Step 3: Bring all terms to one side.* \[ -x^2+5x+6+8 \le 0 \quad \Longrightarrow \quad -x^2+5x+14 \le 0. \] *Step 4: Multiply both sides by \(-1\) (remember to reverse the inequality sign).* \[ x^2-5x-14 \ge 0. \] *Step 5: Factor the quadratic expression (or use the quadratic formula).* Calculate the discriminant: \[ \Delta = (-5)^2-4(1)(-14)=25+56=81. \] Find the roots: \[ x=\frac{5\pm\sqrt{81}}{2}=\frac{5\pm9}{2}. \] Thus, the roots are \[ x=-2 \quad \text{and} \quad x=7. \] *Step 6: Analyze the sign of the quadratic \(x^2-5x-14=(x-7)(x+2)\).* Since the quadratic has a positive leading coefficient, it is nonnegative when \[ x\leq -2 \quad \text{or} \quad x\geq 7. \] *Solution for (a):* \[ x \in (-\infty, -2] \cup [7, \infty). \] *Number line for (a):* Place a closed circle at \(-2\) and at \(7\) with shading to the left of \(-2\) and to the right of \(7\). --- **(b)** We start with the inequality \[ 2x(x+2) < (x+1)(x+3). \] *Step 1: Expand both sides.* Left-hand side: \[ 2x(x+2)=2x^2+4x. \] Right-hand side: \[ (x+1)(x+3)=x^2+4x+3. \] *Step 2: Set up the inequality.* \[ 2x^2+4x < x^2+4x+3. \] *Step 3: Subtract \(x^2+4x\) from both sides.* \[ 2x^2+4x - (x^2+4x) <3 \quad \Longrightarrow \quad x^2<3. \] *Step 4: Solve \(x^2<3\).* Taking square roots and considering both positive and negative solutions gives \[ -\sqrt{3} < x < \sqrt{3}. \] *Solution for (b):* \[ x \in (-\sqrt{3}, \sqrt{3}). \] *Number line for (b):* Draw an open circle at \(-\sqrt{3}\) and at \(\sqrt{3}\) with shading in the interval between them.