What is the solution to the inequality $|(3x+4)/2| - 5 \geq 8$?
Answer (2)
Isolate the absolute value term: ∣ ( 3 x + 4 ) /2∣ g e q 13 .
Split the inequality into two cases: ( 3 x + 4 ) /2 g e q 13 and ( 3 x + 4 ) /2 ≤ − 13 .
Solve each case separately: xg e q 22/3 and x ≤ − 10 .
Combine the solutions: x ≤ − 10 or xg e q 22/3 . The final answer is x ≤ − 10 or xg e q 3 22 .
Explanation
Understanding the Inequality We are given the inequality ∣ ( 3 x + 4 ) /2∣ − 5 ≥ 8 . Our goal is to find all values of x that satisfy this inequality.
Isolating the Absolute Value First, we want to isolate the absolute value term. To do this, we add 5 to both sides of the inequality: ∣ ( 3 x + 4 ) /2∣ − 5 + 5 ≥ 8 + 5 ∣ ( 3 x + 4 ) /2∣ ≥ 13
Considering Two Cases Now, we consider two cases based on the definition of absolute value.
Case 1: The expression inside the absolute value is non-negative, so we have: ( 3 x + 4 ) /2 ≥ 13 Case 2: The expression inside the absolute value is negative, so we have: − ( 3 x + 4 ) /2 ≥ 13 which is equivalent to ( 3 x + 4 ) /2 ≤ − 13
Solving Case 1 Let's solve Case 1: ( 3 x + 4 ) /2 ≥ 13 Multiply both sides by 2: 3 x + 4 ≥ 26 Subtract 4 from both sides: 3 x ≥ 22 Divide both sides by 3: x ≥ 22/3
Solving Case 2 Now, let's solve Case 2: ( 3 x + 4 ) /2 ≤ − 13 Multiply both sides by 2: 3 x + 4 ≤ − 26 Subtract 4 from both sides: 3 x ≤ − 30 Divide both sides by 3: x ≤ − 10
Combining the Solutions Combining the solutions from both cases, we have x ≤ − 10 or x ≥ 22/3 .
Final Answer Therefore, the solution to the inequality is x ≤ − 10 or x ≥ 22/3 .
Examples
Absolute value inequalities can be used in various real-life scenarios, such as determining acceptable ranges in manufacturing or engineering. For example, if a machine part needs to be within a certain tolerance of a specified measurement, an absolute value inequality can be used to define the acceptable range. Suppose a metal rod needs to be 10 cm long, with a tolerance of 0.1 cm. This can be expressed as ∣ x − 10∣ ≤ 0.1 , where x is the actual length of the rod. Solving this inequality gives the acceptable range of lengths for the rod: 9.9 ≤ x ≤ 10.1 .
The solution to the inequality 2 3 x + 4 − 5 ≥ 8 is x ≤ − 10 or x ≥ 3 22 . This means that the values of x that satisfy the inequality are those less than or equal to -10 or greater than or equal to 3 22 .
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