$3 \leqslant 4 x-1<5$
Answer (1)
Add 1 to all parts of the inequality: 3 + 1 l e q s l an t 4 x − 1 + 1 < 5 + 1 , which simplifies to 4 l e q s l an t 4 x < 6 .
Divide all parts of the inequality by 4: 4 4 l e q s l an t 4 4 x < 4 6 , which simplifies to 1 l e q s l an t x < 2 3 .
The solution to the inequality is 1 l e q s l an t x < 2 3 .
In interval notation, the solution is [ 1 , 2 3 ) . [ 1 , 2 3 )
Explanation
Understanding the Inequality We are given the compound inequality 3 l e q s l an t 4 x − 1 < 5 , and our goal is to isolate x to find the solution set.
Adding 1 to All Parts First, we add 1 to all parts of the inequality to start isolating the term with x :
3 + 1 l e q s l an t 4 x − 1 + 1 < 5 + 1 This simplifies to: 4 l e q s l an t 4 x < 6
Dividing by 4 Next, we divide all parts of the inequality by 4 to isolate x :
4 4 l e q s l an t 4 4 x < 4 6 This simplifies to: 1 l e q s l an t x < 2 3
Final Solution Therefore, the solution to the inequality is 1 l e q s l an t x < 2 3 . In interval notation, this is written as [ 1 , 2 3 ) .
Examples
Imagine you are baking a cake and the recipe says you need to keep the oven temperature within a certain range for the cake to bake properly. This range can be expressed as an inequality, just like the one we solved. For instance, if the recipe requires the oven temperature to be between 300°F and 350°F, you need to make sure your oven setting falls within this range to achieve the best results. Similarly, in manufacturing, maintaining precise measurements within specified tolerances is crucial for quality control. Inequalities help define these acceptable ranges, ensuring that products meet the required standards.
