Solve the following inequality and graph the solution.
$6 x+4 \leq 34$
Answer (1)
Subtract 4 from both sides: 6 x + 4 − 4 ≤ 34 − 4 , which simplifies to 6 x ≤ 30 .
Divide both sides by 6: 6 6 x ≤ 6 30 , resulting in x ≤ 5 .
The solution set includes all values of x less than or equal to 5.
The final answer is x ≤ 5 , which can be graphed on a number line with a closed circle at 5 and an arrow extending to the left. x ≤ 5
Explanation
Understanding the Inequality We are given the inequality 6 x + 4 ≤ 34 . Our goal is to isolate x on one side of the inequality to find the solution set.
Subtracting 4 from Both Sides First, we subtract 4 from both sides of the inequality to isolate the term with x : 6 x + 4 − 4 ≤ 34 − 4 6 x ≤ 30
Dividing by 6 Next, we divide both sides of the inequality by 6 to solve for x : 6 6 x ≤ 6 30 x ≤ 5
Solution Set The solution to the inequality is x ≤ 5 . This means that any value of x that is less than or equal to 5 will satisfy the original inequality.
Graphing the Solution To graph the solution, we draw a number line. We place a closed circle (or a filled-in dot) at 5 to indicate that 5 is included in the solution. Then, we draw an arrow extending to the left from 5 to indicate that all numbers less than 5 are also included in the solution.
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, if you're budgeting your expenses, you might want to ensure that your total spending ( x ) plus a fixed cost (like a subscription fee of $4) remains less than or equal to your monthly income ( 34 ) . T hi sc anb ere p rese n t e d a s x + 4 \leq 34$. Solving this inequality helps you determine the maximum amount you can spend while staying within your budget. Similarly, inequalities are used in various fields like engineering, economics, and computer science to model constraints and optimize solutions.
