Before isolating the variable in a messy inequality, two things need to happen first. What are they, and in what order? Think back to the Clean-Up First Strategy, then try applying it to this inequality.

Apply the Clean-Up First Strategy to solve $3(2 x-5)-4 x>7$. Show the simplified expression and the final solution for $x$.

Question

preston51 preston51

30-10-2026
Mathematics

Answered

Before isolating the variable in a messy inequality, two things need to happen first. What are they, and in what order? Think back to the Clean-Up First Strategy, then try applying it to this inequality.

Apply the Clean-Up First Strategy to solve $3(2 x-5)-4 x>7$. Show the simplified expression and the final solution for $x$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-10-30T21:30:49-04:00

The 'Clean-Up First' strategy involves two steps: first, distribute terms to remove parentheses, and second, combine like terms to simplify the expression.
For the inequality $3(2x - 5) - 4x > 7$, distributing the 3 results in $6x - 15 - 4x > 7$.
Combining like terms yields the simplified inequality $2x - 15 > 7$.
Solving this simplified form gives the final answer: $\boxed{x > 11}$.

Explanation

Defining the Clean-Up First Strategy

The 'Clean-Up First' strategy is a systematic approach to solving equations and inequalities by simplifying expressions before isolating variables. The two essential steps, performed in order, are:

Distribution/Expansion: Remove any parentheses by multiplying the constant outside the parentheses by each term inside.
Combining Like Terms: Simplify expressions on each side by grouping terms that have the same variable power or are constants.

We are given the inequality $3(2x - 5) - 4x > 7$. Applying this strategy will allow us to isolate $x$ efficiently.

Applying the Clean-Up Steps

First, we distribute the $3$ across the terms in the parentheses:
\[3(2x) - 3(5) - 4x > 7\]
\[6x - 15 - 4x > 7\]
Next, we combine the like terms on the left-hand side, which are the $x$-terms ($6x$ and $-4x$):
\[(6x - 4x) - 15 > 7\]
\[2x - 15 > 7\]
The expression is now "cleaned up" and simplified into a standard form.

Solving for x

With the inequality simplified to $2x - 15 > 7$, we can now proceed to isolate the variable $x$.
First, add $15$ to both sides of the inequality:
\[2x > 7 + 15\]
\[2x > 22\]
Finally, divide both sides by $2$:
\[x > \frac{22}{2}\]
\[x > 11\]
Thus, the solution to the original inequality is $x > 11$.

Examples

The 'Clean-Up First' strategy is highly applicable in real-world scenarios like financial budgeting. Imagine you have a monthly budget for office supplies, and you need to calculate how many units of a product you can purchase after paying a fixed shipping fee. If your total cost equation is something like $5(x + 2) - 2x = 30$, where $x$ is the number of units and $\$2$ is the cost per unit for handling, you would first distribute the cost and simplify the equation to find out how many units you can actually afford. This ensures you accurately track your spending before making final decisions.