How do you solve the following compound inequality:

\[ 4b + 18 \geq -12b \]
\[ -14 \leq 14 - 5b \]

4 b + 18 ≥ − 12 b − 14 ≤ 14 − 5 b 4 b + 18 ≥ − 12 b − 14 an d − 12 b − 14 ≤ 14 − 5 b 16 b ≥ − 32 an d − 7 b ≤ 28 b ≥ − 2 an d b ≥ 4 S o l u t i o n i s b ∈ [ 4 , + ∞ )

Answered by luana | 2024-06-10

The final solution is where the individual solutions overlap, which is -4 "<=" b ">=" -2.
To solve the compound inequality 4b+18 ">=" -12b -14 "<=" 14 -5b, we need to treat each part of the inequality separately and then combine the solutions.
Step 1: Solve the first inequality
4b + 18 ">=" -12b - 14

Add 12b to both sides: 16b + 18 ">=" -14
Subtract 18 from both sides: 16b ">=" -32
Divide by 16: b ">=" -2

Step 2: Solve the second inequality
-12b - 14 "<=" 14 - 5b

Add 12b to both sides: -14 "<=" 14 + 7b
Subtract 14 from both sides: -28 "<=" 7b
Divide by 7: -4 "<=" b

Step 3: Combine the solutions
The solution to the compound inequality is where the two individual solutions overlap: -4 "<=" b ">=" -2

Answered by Adella123 | 2024-06-25

To solve the compound inequalities separately, we found that b ≥ − 8 9 ​ from the first inequality and b ≤ 5.6 from the second. Combining these results gives the solution b ∈ [ − 1.125 , 5.6 ] .
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Answered by luana | 2024-09-06