Solve the inequalities:
We are tasked with solving two independent linear inequalities for the variable \(x\):
Each inequality represents a linear relationship on a number line. Our goal is to isolate \(x\) in each case to determine the range of values that satisfy the given conditions. Linear inequalities follow similar algebraic rules to linear equations, with the key exception that multiplying or dividing by a negative number reverses the direction of the inequality sign.
To solve \(3 > 2x + 11\), we follow these steps:
\(3 - 11 > 2x\)
\(-8 > 2x\)
\(-8 / 2 > x\)
\(-4 > x\)
This can also be written as \(x < -4\). Any real number less than -4 satisfies this condition.
To solve \(x + 1 \geq 3x - 18\), we follow these steps:
\(1 \geq 2x - 18\)
\(1 + 18 \geq 2x\)
\(19 \geq 2x\)
\(19 / 2 \geq x\)
\(9.5 \geq x\)
This can also be written as \(x \leq 9.5\). Any real number less than or equal to 9.5 satisfies this condition.
We have solved both inequalities independently.
For (a), the solution is \(x < -4\).
For (b), the solution is \(x \leq 9.5\).
These results provide the complete range of values for \(x\) that satisfy each respective condition. The solutions are \(\boxed{x < -4}\) for (a) and \(\boxed{x \leq 9.5}\) for (b).
Linear inequalities are essential in budgeting and resource management. For instance, if a company has a fixed budget of \(\$100\) and they want to purchase items that cost \(\$2\) each, but there is an additional fixed fee of \(\$11\), the inequality \(2x + 11 \leq 100\) helps them find the maximum number of items \(x\) they can afford. Solving such problems allows businesses to stay within their financial limits while planning their operations. Similarly, these inequalities are used in scheduling, where time constraints must be met to complete a project on time.