Solve the first inequality, 2x+5 1.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Solve the first inequality, $2x+5 \ge 1$.
Subtract $5$ from both sides:
$$ 2x \ge 1 - 5 $$
$$ 2x \ge -4 $$
Divide by $2$:
$$ x \ge \frac{-4}{2} $$
$$ x \ge -2 $$
Step 2: Solve the second inequality, $\frac{x+6}{3} < 4$.
Multiply both sides by $3$:
$$ x+6 < 4 \times 3 $$
$$ x+6 < 12 $$
Subtract $6$ from both sides:
$$ x < 12 - 6 $$
$$ x < 6 $$
Step 3: Combine the solutions from both inequalities.
For $x$ to satisfy both inequalities, it must be greater than or equal to $-2$ AND less than $6$.
This can be written as:
$$ -2 \le x < 6 $$
In interval notation, this is:
$$ \boxed{[-2, 6)} $$
What's next?
