Solve the inequality: $4x + 3 > 2x + 11$

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01-11-2026
Mathematics

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Solve the inequality: $4x + 3 > 2x + 11$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T15:23:24-05:00

Subtract $2x$ from both sides to group the $x$-terms: $2x + 3 > 11$.
Subtract $3$ from both sides to isolate the variable term: $2x > 8$.
Divide both sides by $2$ to solve for $x$: $x > 4$.
The solution is the set of all real numbers $x$ such that $x > 4$, denoted by $\boxed{{x > 4}}$.

Explanation

Analysis of the Inequality

We are given the linear inequality $4x + 3 > 2x + 11$. The goal is to isolate the variable $x$ on one side of the inequality to determine the range of values that satisfy it. A linear inequality behaves similarly to a linear equation when performing additions, subtractions, and multiplications or divisions by positive numbers. The primary difference is that the direction of the inequality must be reversed if we multiply or divide by a negative number. Here, we can manipulate the inequality using standard algebraic operations.

Isolating the Variable Term

First, we collect the variable terms on one side by subtracting $2x$ from both sides of the inequality:
\[4x - 2x + 3 > 11\]
\[2x + 3 > 11\]
Next, we isolate the term containing $x$ by subtracting 3 from both sides:
\[2x > 11 - 3\]
\[2x > 8\]

Solving for x

Finally, we solve for $x$ by dividing both sides of the inequality by the positive coefficient 2. Since we are dividing by a positive number, the inequality sign remains unchanged:
\[x > \frac{8}{2}\]
\[x > 4\]
Thus, any value of $x$ that is strictly greater than 4 will satisfy the original inequality.

Examples

Linear inequalities are fundamental in real-world scenarios where you have constraints or thresholds. For example, suppose you are planning an event with a fixed budget. If the catering service charges a base fee of \$@@MATH_TOKEN_23@@ plus \$$4$ per guest, and another service charges a base fee of \$@@MATH_TOKEN_25@@ plus \$$2$ per guest, you would use this inequality to determine how many guests you need to invite for the second option to be cheaper, or conversely, for the first option to be more expensive. Solving $4x + 3 > 2x + 11$ helps you realize that once you have more than 4 guests, the first catering service becomes more expensive than the second.