What value of $x$ is in the solution set of the inequality $-2(3 x+2)>-8 x+6 ?$
Answer (1)
Distribute -2 on the left side: -8x + 6"> − 6 x − 4 > − 8 x + 6 .
Add 8 x to both sides: 6"> 2 x − 4 > 6 .
Add 4 to both sides: 10"> 2 x > 10 .
Divide both sides by 2: 5"> x > 5 . The only option that satisfies this is 6 .
Explanation
Understanding the Problem We are given the inequality -8x + 6"> − 2 ( 3 x + 2 ) > − 8 x + 6 , and we need to find which of the given values for x satisfies this inequality.
Distributing First, let's solve the inequality for x . Distribute the − 2 on the left side of the inequality: -8x + 6"> − 2 ( 3 x + 2 ) > − 8 x + 6
-8x + 6"> − 6 x − 4 > − 8 x + 6
Adding 8x to Both Sides Next, add 8 x to both sides of the inequality: -8x + 6 + 8x"> − 6 x − 4 + 8 x > − 8 x + 6 + 8 x
6"> 2 x − 4 > 6
Adding 4 to Both Sides Now, add 4 to both sides of the inequality: 6 + 4"> 2 x − 4 + 4 > 6 + 4
10"> 2 x > 10
Dividing by 2 Finally, divide both sides of the inequality by 2 : \frac{10}{2}"> 2 2 x > 2 10
5"> x > 5
Checking the Options Now we need to check which of the given options satisfy the inequality 5"> x > 5 . The options are − 6 , − 5 , 5 , 6 .
For x = − 6 , 5"> − 6 > 5 is false.
For x = − 5 , 5"> − 5 > 5 is false.
For x = 5 , 5"> 5 > 5 is false.
For x = 6 , 5"> 6 > 5 is true.
Therefore, only x = 6 satisfies the inequality.
Final Answer The value of x that is in the solution set of the inequality -8x + 6"> − 2 ( 3 x + 2 ) > − 8 x + 6 is 6 .
Examples
Inequalities are used in various real-life situations, such as determining the range of acceptable values for a variable. For example, a company might use an inequality to determine the minimum number of products they need to sell to make a profit. If the profit P is given by P = 5 x − 1000 , where x is the number of products sold, the company wants to find the minimum x such that 0"> P > 0 . Solving 0"> 5 x − 1000 > 0 gives 200"> x > 200 . Thus, the company needs to sell more than 200 products to make a profit.
