To solve the system of inequalities [tex]\( 4x \geq -16 \)[/tex] and [tex]\( x + 3 < 8 \)[/tex], let's break it down step by step.
### Step 1: Solving the inequality [tex]\( 4x \geq -16 \)[/tex]
1. Start with the inequality:
[tex]\[
4x \geq -16
\][/tex]
2. To isolate [tex]\( x \)[/tex], we need to divide both sides of the inequality by 4:
[tex]\[
x \geq \frac{-16}{4}
\][/tex]
3. Simplify the right side:
[tex]\[
x \geq -4
\][/tex]
So, the solution to the first inequality is [tex]\( x \geq -4 \)[/tex].
### Step 2: Solving the inequality [tex]\( x + 3 < 8 \)[/tex]
1. Start with the inequality:
[tex]\[
x + 3 < 8
\][/tex]
2. To isolate [tex]\( x \)[/tex], subtract 3 from both sides:
[tex]\[
x < 8 - 3
\][/tex]
3. Simplify the right side:
[tex]\[
x < 5
\][/tex]
So, the solution to the second inequality is [tex]\( x < 5 \)[/tex].
### Step 3: Combining the solutions
We now have two separate inequalities:
1. [tex]\( x \geq -4 \)[/tex]
2. [tex]\( x < 5 \)[/tex]
To find the values of [tex]\( x \)[/tex] that satisfy both inequalities, we need to find the intersection of these solution sets.
- The first inequality [tex]\( x \geq -4 \)[/tex] means [tex]\( x \)[/tex] can be any number greater than or equal to -4.
- The second inequality [tex]\( x < 5 \)[/tex] means [tex]\( x \)[/tex] can be any number less than 5.
The intersection of [tex]\( x \geq -4 \)[/tex] and [tex]\( x < 5 \)[/tex] is given by:
[tex]\[
-4 \leq x < 5
\][/tex]
### Conclusion:
The solution to the system of inequalities is:
[tex]\[
-4 \leq x < 5
\][/tex]
This means [tex]\( x \)[/tex] can take any value within the interval [tex]\([-4, 5)\)[/tex].
