Solve the linear inequality 2 − x < 1 + bx, expand (2x + 1)2, and find the coefficient of q in the expansion of (2q + 1)(1 - 2q).

here are the solutions for the requested problems.

:

Step 1: The problem asks to solve the linear inequality $2 - x < 1 + bx$.

Step 2: Gather terms involving $x$ on one side and constant terms on the other side.

$$2 - 1 < bx + x$$ $$1 < x(b + 1)$$

Step 3: To isolate $x$, divide both sides by $(b+1)$. The direction of the inequality depends on the sign of $(b+1)$.

• If $b+1 > 0$ (i.e., $b > -1$):

$$x > \frac{1}{b+1}$$

• If $b+1 < 0$ (i.e., $b < -1$):

$$x < \frac{1}{b+1}$$

• If $b+1 = 0$ (i.e., $b = -1$): The inequality becomes $1 < x(0)$, which simplifies to $1 < 0$. This is a false statement, meaning there is no solution for $x$ when $b = -1$.

The solution depends on the value of $b$:
$\boxed{x > \frac{1}{b+1} \text{ if } b > -1}$
$\boxed{x < \frac{1}{b+1} \text{ if } b < -1}$
$\boxed{\text{No solution if } b = -1}$

:

Step 1: The problem asks to expand $(2x + 1)^2$.

Step 2: Use the algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$. Here, $a = 2x$ and $b = 1$.

Step 3: Substitute these values into the identity:

$$(2x + 1)^2 = (2x)^2 + 2(2x)(1) + (1)^2$$

Step 4: Simplify each term:

$$= 4x^2 + 4x + 1$$

The expanded form is $\boxed{4x^2 + 4x + 1}$.

:

Step 1: The problem asks for the coefficient of $q$ in the expansion of $(2q + 1)(1 - 2q)$.

Step 2: Expand the expression using the distributive property (FOIL method):

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