Example: Solving and Graphing $8p + 3(p - 12) > 7p - 28$

Example: Solving and Graphing $9y + 2(y + 6) > 5y - 24$

Example: Solving and Graphing $6u + 8(u - 1) > 10u + 32$

Example: Solving and Graphing $8x - 2(5 - x) < 4(x + 9) + 6x$

Example: Solving and Graphing $4b - 3(3 - b) > 5(b - 6) + 2b$

Readiness Quiz: Solving $4x + 3 > 23$

Solving the Linear Inequality $2a < 5a + 12$

A logistics coordinator is using the inequality ${}-2x + 400 \leq 100$ to determine the maximum number of additional crates, $x$, that can be added to a shipment without exceeding a weight limit. To isolate $x$ in the final step of the solution, the coordinator must divide both sides by -2. Which rule must be recalled when performing this operation?

A production manager is solving a multi-step linear inequality to determine the minimum number of units a factory must produce to stay within a specific budget. Arrange the following steps in the standard order typically used to solve a multi-step linear inequality for a variable, such as ${}x$.

A procurement officer is solving the inequality $-5x + 1200 \leq 400$ to stay within a quarterly budget for office supplies, where $x$ represents the number of units ordered. After simplifying the inequality to $-5x \leq -800$, the officer must divide both sides by $-5$ to solve for $x$. True or False: In this final step, the inequality sign $\leq$ must be reversed to $\geq$ to maintain the correct solution.

Simplifying Multi-Step Inequalities

A business operations manager is solving a multi-step linear inequality to determine the maximum number of safety inspections that can be performed within a fixed quarterly budget. Match each algebraic property or rule to the correct description of its role in the solution process.

Solving the Linear Inequality $2a < 5a + 12$

As a logistics coordinator, you use the expression ${}4x + 3$ to determine the total shipping cost in dollars, where $x$ is the number of expedited packages. If the total cost exceeds 23 dollars, the system flags the order for review, modeled by the inequality ${}4x + 3 > 23$. Recalling the standard algebraic sequence to isolate a variable, what is the correct solution for $x$?

You are a facility manager at a corporate campus calculating the threshold for extra maintenance hours ($x$) during a major renovation project. Your budget requires that the total labor cost, represented by the expression ${}4x + 3$, must exceed 23 dollars per unit of work to trigger a secondary funding source, modeled by the inequality ${}4x + 3 > 23$. To determine the point at which this funding is activated, you must solve for $x$. Arrange the following steps in the correct algebraic sequence to isolate the variable.

A telecommunications billing analyst is auditing an automated system that flags accounts for excessive data usage. The system uses the inequality ${}4x + 3 > 23$ to determine when to trigger a surcharge alert, where $x$ represents the number of additional gigabytes used. To verify the audit logic, match each component of the solution process with its correct algebraic description or result.

Operational Expense Threshold

A procurement analyst uses the inequality ${}4x + 3 > 23$ to determine the minimum number of units ($x$) required to qualify for a corporate bulk discount. After performing the necessary algebraic steps to isolate $x$, the analyst finds that the number of units must be greater than ____.