Find the complete set of values of [tex]$x$[/tex] which satisfy the inequality
[tex]$\frac{1}{2}(2 x+3)-\frac{2}{3}(x+1)\ \textless \ 2 x$[/tex]
A. [tex]$x\ \textgreater \ \frac{1}{2}$[/tex]
B. [tex]$x\ \textgreater \ \frac{5}{14}$[/tex]
C. [tex]$x\ \textgreater \ \frac{7}{8}$[/tex]
D. [tex]$x\ \textgreater \ \frac{5}{4}$[/tex]
Answer (2)
Expand the left side of the inequality.
Combine like terms.
Isolate x on one side.
Solve for x: \frac{1}{2}"> x > 2 1 .
Explanation
Understanding the Problem We are given the inequality 2 1 ( 2 x + 3 ) − 3 2 ( x + 1 ) < 2 x . Our goal is to find all values of x that satisfy this inequality.
Expanding the Inequality First, we expand the left side of the inequality: 2 1 ( 2 x + 3 ) − 3 2 ( x + 1 ) < 2 x x + 2 3 − 3 2 x − 3 2 < 2 x
Combining Like Terms Next, we combine like terms on the left side: x − 3 2 x + 2 3 − 3 2 < 2 x 3 3 x − 3 2 x + 6 9 − 6 4 < 2 x 3 1 x + 6 5 < 2 x
Isolating x Now, we want to isolate x . Subtract 3 1 x from both sides: 6 5 < 2 x − 3 1 x 6 5 < 3 6 x − 3 1 x 6 5 < 3 5 x
Solving for x Finally, we solve for x by multiplying both sides by 5 3 : 5 3 × 6 5 < 5 3 × 3 5 x 30 15 < x 2 1 < x So, \frac{1}{2}"> x > 2 1 .
Examples
Understanding inequalities is crucial in various real-world scenarios, such as determining the range of values for a variable in optimization problems. For instance, if you're trying to maximize profit within a certain budget, inequalities help define the feasible region where you can operate. In this case, solving the inequality helps us find the range of x values that satisfy the given condition, which can be applied to resource allocation or decision-making processes.
The solution to the inequality 2 1 ( 2 x + 3 ) − 3 2 ( x + 1 ) < 2 x leads to the conclusion that \frac{1}{2}"> x > 2 1 . Therefore, the correct choice is option A: \frac{1}{2}"> x > 2 1 .
;
