Select the correct answer.
Irene made a mistake when she solved this inequality:
Step 1: $-6(x+3)+10<-2$
Step 2: $-6 x-18+10<-2$
Step 3: $-6 x-8<-2$
Step 4: $-6 x<6$
Step 5: $x<-1$
Between which two consecutive steps did Irene make a mistake?
A. step 1 and step 2
B. step 2 and step 3
C. step 3 and step 4
D. step 4 and step 5
Answer (1)
Distribute and simplify the initial inequality.
Isolate the term with x .
Divide by the coefficient of x , remembering to flip the inequality sign if dividing by a negative number.
Identify that the mistake was in the last step where the inequality sign was not flipped, so the answer is D .
Explanation
Identifying the Mistake Let's analyze Irene's steps to identify the mistake in solving the inequality.
Step 1: − 6 ( x + 3 ) + 10 < − 2 This is the initial inequality. Step 2: − 6 x − 18 + 10 < − 2 Here, the distributive property is applied: − 6 ∗ x + ( − 6 ) ∗ 3 = − 6 x − 18 . This step is correct. Step 3: − 6 x − 8 < − 2 Combining the constants: − 18 + 10 = − 8 . This step is also correct. Step 4: − 6 x < 6 Adding 8 to both sides of the inequality: − 6 x − 8 + 8 < − 2 + 8 , which simplifies to − 6 x < 6 . This step is correct. Step 5: x < − 1 Dividing both sides by -6. When dividing or multiplying an inequality by a negative number, the inequality sign must be flipped. Irene did not flip the inequality sign. Therefore, this step is incorrect. The correct step should be -1"> x > − 1 .
The mistake occurred when Irene divided both sides of the inequality by -6 in Step 4. She forgot to flip the inequality sign.
Locating the Mistake The mistake occurred between Step 4 and Step 5 because Irene forgot to flip the inequality sign when dividing by a negative number.
Final Answer Therefore, the correct answer is D. step 4 and step 5
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 inequalities to determine the minimum and maximum number of products they need to sell to make a profit. If the profit, P , depends on the number of products sold, x , and the cost and revenue functions are known, the company can set up an inequality like 0"> P > 0 to find the range of x that results in a positive profit. Solving this inequality helps the company make informed decisions about production and sales targets.
