Solve the inequality $11-2 x<3$. Graph the solution set.
In set notation, the solution is \{x | $\square$ \}. (Type an inequality.)
Answer (1)
Subtract 11 from both sides: − 2 x < − 8 .
Divide both sides by -2 (and flip the inequality sign): 4"> x > 4 .
Express the solution in set notation: 4\}"> { x ∣ x > 4 } .
The solution to the inequality is 4}"> x > 4 .
Explanation
Understanding the Inequality We are given the inequality 11 − 2 x < 3 . Our goal is to isolate x on one side of the inequality to find the solution set.
Subtracting 11 from Both Sides First, we subtract 11 from both sides of the inequality: 11 − 2 x − 11 < 3 − 11 This simplifies to: − 2 x < − 8
Dividing by -2 and Flipping the Inequality Next, we divide both sides of the inequality by -2. Remember that when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign. So, we have: \frac{-8}{-2}"> − 2 − 2 x > − 2 − 8
This simplifies to: 4"> x > 4
Expressing the Solution in Set Notation Now, we express the solution in set notation. The solution set includes all values of x that are greater than 4. Therefore, the solution in set notation is: 4\}"> { x ∣ x > 4 }
Final Answer The solution to the inequality 11 − 2 x < 3 is 4"> x > 4 . In set notation, this is represented as 4\}"> { x ∣ x > 4 } .
Examples
Understanding inequalities is crucial in various real-life situations, such as determining budget constraints. For instance, if you have a budget of $50 and want to buy items costing $5 each, the inequality 5 x < 50 helps you determine the maximum number of items you can purchase. Solving this inequality, you find that x < 10 , meaning you can buy a maximum of 9 items to stay within your budget. This concept extends to more complex scenarios like investment planning, where you might use inequalities to model potential returns and risks.
