Question

To which of the following inequalities is the number 11 a solution? A. \(3 x-5<2(2-x)\) B. \(7 x-3 \geq 7-3 x\) C. \(5(x-2) \leq 3(x+1)\) D. \(4(x-8)>2 x+5\)

Short answer

The number 11 is a solution for inequality B.

Step-by-step solution

- Evaluate Inequality A

Plug in 11 for x and simplify the inequality \[3(11) - 5 < 2(2 - 11)\]\[33 - 5 < 2(-9)\]\[28 < -18\]This simplification is not true, so 11 is not a solution for inequality A.

- Evaluate Inequality B

Plug in 11 for x and simplify the inequality \[7(11) - 3 \geq 7 - 3(11)\]\[77 - 3 \geq 7 - 33\]\[74 \geq -26\]This simplification is true, so 11 is a solution for inequality B.

- Evaluate Inequality C

Plug in 11 for x and simplify the inequality \[5(11 - 2) \leq 3(11 + 1)\]\[5(9) \leq 3(12)\]\[45 \leq 36\]This simplification is not true, so 11 is not a solution for inequality C.

- Evaluate Inequality D

Plug in 11 for x and simplify the inequality \[4(11 - 8) > 2(11) + 5\]\[4(3) > 22 + 5\]\[12 > 27\]This simplification is not true, so 11 is not a solution for inequality D.

Key concepts

Solution to Inequalities

When facing inequalities in math, we aim to find the range of values that satisfy the given conditions. A solution to an inequality is any value of the variable that makes the inequality true.
For example, in the exercise, we tested whether the number 11 could make each inequality true. By substituting 11 for the variable, we evaluated each inequality step by step:

• For inequality A, substituting 11 resulted in a false statement, so it wasn't a solution.
• For inequality B, substituting 11 provided a true statement, indicating 11 is a solution.
• Inequalities C and D, when tested with 11, resulted in false statements, showing that 11 is not a solution for these inequalities.

By checking each inequality through substitution and simplification, we determine if the number meets the conditions of each inequality.

Solving Inequalities

To solve inequalities, you follow similar steps to solving equations, with a few additional rules. The key idea is to isolate the variable on one side of the inequality. Here are the main steps:

• First, simplify both sides of the inequality if needed. For example, combine like terms.
• Next, isolate the variable by using inverse operations like addition, subtraction, multiplication, or division.
• Remember, when you multiply or divide both sides of an inequality by a negative number, the inequality symbol must be reversed.

Let's apply these ideas to our exercise. For instance, in inequality B: \[7 x - 3 \geq 7 - 3 x\] we simplified and solved it by substituting 11 for x, giving \[74 \geq -26\text{, meaning the statement is true and } 11 \text{ is a solution.}\]
Understanding and following these steps helps you solve most inequalities effectively.

Substitution Method

The substitution method involves replacing the variable with a given value to verify if it satisfies the inequality. This method is especially useful for checking specific solutions.
Here’s how to use substitution step by step:

• Identify the given value. In our exercise, this value was 11.
• Substitute this value into the inequality wherever the variable appears.
• Simplify the resulting expression.
• Check if the resulting statement is true or false to conclude if the given value is a solution.

Let's look at inequality C for illustration:
Substituting 11 for x gives: \[5(11-2) \leq 3(11+1)\] which simplifies to \[45 \leq 36\]. This is a false statement, meaning 11 is not a solution for inequality C.
This method helps quickly verify potential solutions to complex inequalities, ensuring you efficiently identify whether a specific value meets all conditions.