Solving inequalities number line

Question

So I am currently doing inequality problems and i have the answer to the question but am unsure on how to put that onto a number line.

The question is $-7+b≤6b-2<7b+1$.

The answers i got for this are $b≥-1$ and $b>-3$ and am wondering on if the answer is correct and how to put this onto a number line help would be greatly appreciated.

Answer 1

$\color{red}{-7+b} ≤ \color{green}{6b-2} < \color{blue}{7b+1}$ means two inequalities:

$(1)$ $\color{red}{-7+b} ≤ \color{green}{6b-2}$.

Solving $(1)$ gives $-1 \leq b$.

$(2)$ $ \color{green}{6b-2} < \color{blue}{7b+1}$.

Solving $(2)$ gives $-3 < b$.

In total we have $b \geq -1 >-3$.

$b \geq -1$ is enough.

Answer 2

Go step wise!

First solve $-7+b\le 6b-2\implies -7+2\le 6b-b\implies b\ge-1$

Now next $6b-2< 7b+1\implies -2-1< 7b-6b\implies b> -3$

So yes, you are right. Now to proceed note that $-1>-3$. Thus $b\ge -1$ staisfies both conditions.

Answer 3

I like the following way: $$-7+b\leq 6b-2<7b+1$$ it's $$-5\leq 5b<6b+3$$ or $$-1\leq b$$ and $$b>-3$$ or $$b\geq-1.$$