From a + 2b = 3 we get that b = (3 - a)/2.

Substitute b = (3 - a)/2 into a + b < 0 to get a + (3 - a)/2 < 0. This simplifies in a < -3.

Now, if a < -3, then a is also less than -2.

Answer: C.

given a+b<0
so sum is -ve and a+2b=3
let a = -5 and b = 4
option C a<-2 is correct

Here's the step-by-step solution:

1. From the second equation, we can express a in terms of b:
a + 2b = 3 implies a = 3 - 2b

2. Substitute a = 3 - 2b into the inequality a + b < 0:

(3 - 2b) + b < 0

Simplify the inequality:

3 - b < 0 \implies 3 < b \implies b > 3


3. Since b > 3, substitute back into a = 3 - 2b:

a = 3 - 2b

Since b > 3, we can infer that 2b > 6. Thus:

a = 3 - 2b < 3 - 6 = -3


Therefore, a < -3.

Among the given options, the one that is consistent with a < -3 is answer C a < -2

I'm confused here.

IF a+2b=3 , why can't b= (3-a)/2 ? Even with a = 3-2b, when a = -3, then a+b = 0, not < 0

You can prove a < -2 when you plug it in, because -2 = 3-2b -> -5 = -2b -> b=2.5 ; -2+2.5 > 0 , so a < -2
But on that same note, when a <-3, -3 = 3-2b -> -6=-2b -> b=3 ; -3+3 = 0 , which does not work for a+b<0 ; So a can't equal -3

So how can we evaluate that a<-2 if -3 doesn't work?
a < -4 does work because -4 = 3-2b -> -7=-2b -> b=7/2, or 3.5 ; -4+3.5 -> -.5<0 which works

I'm just confused

To solve this, we need to express the relationship entirely in terms of $a$. We are given two pieces of information:
1. $\(a+b<0\)$
2. $\(a+2 b=3\)$

Step 1: Solve the equation for $b$ From the second equation, we isolate $b$ :

$$
\(\begin{aligned}
& a+2 b=3 \\
& 2 b=3-a \\
& b=\frac{3-a}{2}
\end{aligned}\)
$$


Step 2: Substitute $b$ into the inequality Now, substitute this expression for $b$ into the first inequality:

$$
\(a+\left(\frac{3-a}{2}\right)<0\)
$$


Step 3: Solve for $a$ Multiply the entire inequality by 2 to clear the fraction:

$$
\(2 a+(3-a)<0\)
$$


Combine like terms:

$$
\(a+3<0\)
$$


$$
\(a<-3\)
$$


Evaluating the Options
We have determined that $a$ must be less than -3 . Now we look at which of the provided statements must be true if $\(a<-3\)$ :
- $\(a>4\)$ : False. $a$ is a negative number.
- $\(a<-4\)$ : Not necessarily true. For example, if $\(a=-3.5\)$, the original conditions are met, but $\(a<-4\)$ is false.
- $\(a<-2\)$ : True. If a number is less than -3 , it is mathematically guaranteed to be less than -2 . Therefore, this statement must be true.
- $\(2<a<4\)$ : False. $a$ is a negative number.