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': x ≤ 2 or x ≥ 7 .
Express the solution in interval notation: ( − ∞ , 2 ] ∪ [ 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 the solution will include all values of x that satisfy either inequality.
Solving the First Inequality First, let's solve the inequality 6 x + 8 ≤ 20 . We want to isolate x on one side of the inequality.
Isolating the x term Subtract 8 from both sides of the inequality: 6 x + 8 − 8 ≤ 20 − 8 6 x ≤ 12
Solving for x Divide both sides by 6: 6 6 x ≤ 6 12 x ≤ 2
Solving the Second Inequality Now, let's solve the second inequality 5 + 4 x ≥ 33 . Again, we want to isolate x .
Isolating the x term Subtract 5 from both sides of the inequality: 5 + 4 x − 5 ≥ 33 − 5 4 x ≥ 28
Solving for x Divide both sides by 4: 4 4 x ≥ 4 28 x ≥ 7
Combining the Solutions The solution to the compound inequality is x ≤ 2 or x ≥ 7 . In interval notation, this is ( − ∞ , 2 ] ∪ [ 7 , ∞ ) .
Examples
Imagine you're planning a road trip and need to decide which routes to take based on speed limits. One route has a maximum speed of 60 mph for a portion of the trip, while another route has a minimum speed of 70 mph for a different segment. Solving inequalities helps you determine which speeds are acceptable on each route, ensuring you stay within the legal limits and arrive safely. This type of problem applies to various scenarios, such as managing budgets, setting performance goals, or determining acceptable ranges in scientific experiments.
