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Magazine Chapter 9: Problem 85
Solve the inequality \(3(a+2) \leq 2(3 a+4)\).
Short Answer
Expert verified
Answer: The solution set is [-2/3, +鈭�).
Step by step solution
01
Apply the Distributive Property
First, we need to distribute the numbers outside the parentheses on both sides of the inequality: \(3(a+2) \leq 2(3a+4)\). To do this, multiply the numbers outside the parentheses with the terms inside the parentheses:
\(3 \times a + 3\times2 \leq 2 \times 3a + 2 \times 4\)
This simplifies to:
\(3a+6 \leq 6a+8\)
02
Move Variables to One Side of the Inequality
In order to isolate the variable a, we will move all a terms to one side of the inequality. Let's subtract 3a from both sides to move the a terms to the right side:
\(3a+6-3a \leq 6a+8-3a\)
This simplifies to:
\(6 \leq 3a+8\)
03
Move Constants to the Opposite Side of the Inequality
Now we will move all constant terms to the left side of the inequality. To do this, we will subtract 8 from both sides:
\(6-8 \leq 3a+8-8\)
This simplifies to:
\(-2 \leq 3a\)
04
Solve for the Variable
Now that we have isolated a on the right side of the inequality, we need to solve for it by dividing both sides by 3:
\(\frac{-2}{3}\leq\frac{3a}{3}\)
This simplifies to:
\(\frac{-2}{3}\leq a\)
05
Write the Answer in Interval Notation
The answer \(\frac{-2}{3}\leq a\) can be written as an interval using the notation [-2/3, +鈭�). This means that the solution set for the inequality \(3(a+2) \leq 2(3a+4)\) is all values of a greater than or equal to -2/3.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributive Property
In order to solve algebraic inequalities, the Distributive Property is often a crucial first step. The Distributive Property helps in simplifying an expression by eliminating parentheses. It states that multiplying a number by a sum is the same as doing each multiplication separately.
For example, in the inequality \(3(a+2) \leq 2(3a+4)\), you apply the Distributive Property as follows:
For example, in the inequality \(3(a+2) \leq 2(3a+4)\), you apply the Distributive Property as follows:
- Multiply \(3\) by \(a\) and \(2\), which gives \(3a + 6\).
- Similarly, multiply \(2\) by \(3a\) and \(4\), which results in \(6a + 8\).
Solving Inequalities
Solving inequalities follows a process similar to solving equations, with a few key differences. Once the expression is simplified using the Distributive Property, your next goal is to isolate the variable involved.
For the inequality \(3a + 6 \leq 6a + 8\):
For the inequality \(3a + 6 \leq 6a + 8\):
- First, get all terms with \(a\) on one side. Subtract \(3a\) from both sides to simplify to \(6 \leq 3a + 8\).
- Next, move the constants to the opposite side by subtracting \(8\) from both sides, leading to \(-2 \leq 3a\).
- Finally, isolate \(a\) by dividing each side by \(3\), resulting in \(\frac{-2}{3} \leq a\).
Interval Notation
After solving an inequality, it's often useful to express the solution in a format called interval notation. This notation provides a clear way to represent the range of values that can satisfy the inequality.
For \(\frac{-2}{3} \leq a\), you write the solution as the interval \([-\frac{2}{3}, +\infty)\). This tells us:
For \(\frac{-2}{3} \leq a\), you write the solution as the interval \([-\frac{2}{3}, +\infty)\). This tells us:
- The interval starts at \(-\frac{2}{3}\), and brackets \([]\) indicate that \(-\frac{2}{3}\) is included in the solution.
- The interval extends to \(+\infty\), the parenthesis \()\) indicates that it never actually reaches infinity, as infinity is not a number that can be reached or a specific value.
