$4x - 3 \geq 2x + 5$
Answer (1)
Subtract 2 x from both sides: 2 x − 3 g e q s l an t 5 .
Add 3 to both sides: 2 xg e q s l an t 8 .
Divide both sides by 2 : xg e q s l an t 4 .
The solution is xg e q s l an t 4 , which in interval notation is [ 4 , ∞ ) .
Explanation
Understanding the Inequality We are given the inequality 4 x − 3 g e q s l an t 2 x + 5 . Our goal is to isolate x on one side of the inequality to find the solution set.
Subtracting 2x First, we subtract 2 x from both sides of the inequality: 4 x − 3 − 2 xg e q s l an t 2 x + 5 − 2 x This simplifies to: 2 x − 3 g e q s l an t 5
Adding 3 Next, we add 3 to both sides of the inequality: 2 x − 3 + 3 g e q s l an t 5 + 3 This simplifies to: 2 xg e q s l an t 8
Dividing by 2 Finally, we divide both sides of the inequality by 2 :
2 2 x g e q s l an t 2 8 This simplifies to: xg e q s l an t 4
Final Solution The solution to the inequality is xg e q s l an t 4 . In interval notation, this is [ 4 , ∞ ) .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, consider a budget constraint: if you have $20 to spend on snacks and each snack costs $2, the inequality 2 x ≤ 20 (where x is the number of snacks) helps you determine the maximum number of snacks you can buy. Solving this gives x ≤ 10 , meaning you can buy at most 10 snacks. Inequalities are also used in setting speed limits, determining safe loads, and calculating tolerance ranges in manufacturing.
