Solve the inequality.
$3(x-2)+4<2 x+9$
We are given the linear inequality \(3(x - 2) + 4 < 2x + 9\). Our goal is to find the range of all real numbers \(x\) that satisfy this condition. The inequality involves basic algebraic operations including distribution, combining like terms, and isolating the variable \(x\). We will simplify both sides systematically to isolate \(x\).
First, we distribute the \(3\) across the terms in the parentheses:
\[3(x - 2) = 3x - 6\]
Substituting this back into the inequality, we get:
\[3x - 6 + 4 < 2x + 9\]
Now, we simplify the left-hand side by combining the constants \(-6\) and \(4\):
\[-6 + 4 = -2\]
So the inequality becomes:
\[3x - 2 < 2x + 9\]
To isolate the variable \(x\), we first subtract \(2x\) from both sides of the inequality:
\[3x - 2x - 2 < 2x - 2x + 9\]
\[x - 2 < 9\]
Next, we add \(2\) to both sides to solve for \(x\):
\[x - 2 + 2 < 9 + 2\]
\[x < 11\]
Thus, the solution to the inequality is all real numbers \(x\) such that \(x < 11\). In interval notation, this is \((-\infty, 11)\).
Linear inequalities are used in many real-world scenarios, such as budgeting or resource allocation. For example, if you have a budget of \(\$11\) and you want to know how many hours you can rent a bicycle that costs \(\$3\) per hour plus a flat fee of \(\$2\) for equipment insurance, plus an additional \(\$4\) for a helmet rental, the cost expression is \(3x + 2 + 4\). To stay within your budget, you would set up the inequality \(3x + 6 \leq 11\), which helps you determine the maximum time you can enjoy the bike. This simple algebraic process helps ensure you never exceed your financial limits.