Question

question
solve the following inequality algebraically.
$-3x^2 - 35 \geq -27x + 7$

Explanation:

Step1: Rearrange to standard form

Bring all terms to left side:
$$-3x^2 - 35 + 27x - 7 \geq 0$$
Simplify:
$$-3x^2 + 27x - 42 \geq 0$$
Multiply by $-1$ (reverse inequality):
$$3x^2 - 27x + 42 \leq 0$$
Divide by 3:
$$x^2 - 9x + 14 \leq 0$$

Step2: Factor the quadratic

Find two numbers: 2 and 7 (sum=9, product=14):
$$(x - 2)(x - 7) \leq 0$$

Step3: Find critical points

Set factors to 0:
$x - 2 = 0 \implies x=2$
$x - 7 = 0 \implies x=7$

Step4: Test intervals

• For $x < 2$: $(x-2)(x-7) > 0$ (positive, not valid)
• For $2 < x < 7$: $(x-2)(x-7) < 0$ (negative, valid)
• For $x > 7$: $(x-2)(x-7) > 0$ (positive, not valid)

Include critical points (inequality is $\leq$)

Answer:

$2 \leq x \leq 7$ (or the interval $[2, 7]$)