Solve $6 x+8 \leq 20$ or $5+4 x \geq 33$
Answer (1)
Solve the first inequality 6 x + 8 ≤ 20 , which gives x ≤ 2 .
Solve the second inequality 5 + 4 x ≥ 33 , which gives x ≥ 7 .
Combine the solutions using 'or', resulting in x ≤ 2 or x ≥ 7 .
The solution to the compound inequality is x ≤ 2 or x ≥ 7 , which can be written as x ≤ 2 or x ≥ 7 .
Explanation
Understanding the Problem We are given the compound inequality 6 x + 8 ≤ 20 or 5 + 4 x ≥ 33 . We need to solve each inequality separately and then combine the solutions using the 'or' condition, which means we take the union of the solution sets.
Solving the First Inequality First, let's solve the inequality 6 x + 8 ≤ 20 .
Subtract 8 from both sides: 6 x + 8 − 8 ≤ 20 − 8 6 x ≤ 12 Divide both sides by 6: 6 6 x ≤ 6 12 x ≤ 2
Solving the Second Inequality Now, let's solve the inequality 5 + 4 x ≥ 33 .
Subtract 5 from both sides: 5 + 4 x − 5 ≥ 33 − 5 4 x ≥ 28 Divide both sides by 4: 4 4 x ≥ 4 28 x ≥ 7
Combining the Solutions The solution to the compound inequality is the union of the solutions to the individual inequalities. So, we have x ≤ 2 or x ≥ 7 . This means x can be any number less than or equal to 2, or any number greater than or equal to 7.
Examples
Understanding inequalities is crucial in various real-life scenarios, such as budgeting and resource allocation. For instance, if you have a limited budget for groceries and want to ensure you have enough money for essential items while also saving for leisure activities, you can use inequalities to determine how much to spend on each category. Similarly, in manufacturing, inequalities help determine the optimal production levels to maximize profit while staying within resource constraints. This problem demonstrates how to solve compound inequalities, which can be applied to more complex optimization problems in economics, engineering, and other fields.
