Solve the inequality. Graph the solution set and write it in interval notation.
1/3 * (x + 3) < 1/7 * (3x + 4)
Write the answer in interval notation.
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(Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)
To solve the inequality 1/3 * (x + 3) < 1/7 * (3x + 4), we need to find the values of x that satisfy the inequality.
Let's begin by simplifying the inequality:
1/3 * (x + 3) < 1/7 * (3x + 4)
Multiplying both sides of the inequality by 21 (the least common denominator of 3 and 7) to eliminate fractions, we get:
7(x + 3) < 3(3x + 4)
Expanding both sides, we have:
7x + 21 < 9x + 12
Subtracting 9x from both sides and subtracting 21 from both sides, we get:
7 - 9x < 12 - 21
-2x < -9
Dividing both sides of the inequality by -2 (remembering to flip the inequality sign when dividing by a negative number), we have:
x > -9 / -2
x > 9/2
So the solution to the inequality 1/3 * (x + 3) < 1/7 * (3x + 4) is x > 9/2.
Graphically, the solution set is all real numbers greater than 9/2 represented by a shaded line extending to the right on the number line.
In interval notation, the solution is (9/2, ∞).
To solve the inequality 1/3 * (x + 3) < 1/7 * (3x + 4), we can start by distributing the fractions:
(x + 3)/3 < (3x + 4)/7
Next, we can multiply both sides of the inequality by 21 (the least common multiple of 3 and 7) to eliminate the fractions:
21 * [(x + 3)/3] < 21 * [(3x + 4)/7]
7(x + 3) < 3(3x + 4)
7x + 21 < 9x + 12
Now, let's isolate the variable by subtracting 7x and 12 from both sides:
7x - 7x + 21 - 12 < 9x - 7x + 12 - 12
9 < 2x
Finally, dividing both sides by 2, we get:
9/2 < x
In interval notation, the solution set is (9/2, infinity).
To solve the inequality, we will start by distributing the fractions on both sides and then simplifying the expression:
1/3 * (x + 3) < 1/7 * (3x + 4)
Multiply both sides of the inequality by the least common denominator (LCD) to eliminate the fractions. The LCD for 3 and 7 is 21:
21 * (1/3 * (x + 3)) < 21 * (1/7 * (3x + 4))
7 * (x + 3) < 3 * (3x + 4)
Now, distribute and simplify further:
7x + 21 < 9x + 12
Subtract 7x from both sides:
21 < 2x + 12
Subtract 12 from both sides:
9 < 2x
Divide both sides by 2:
4.5 < x
The solution to the inequality is x > 4.5. Now let's graph the solution set and write it in interval notation.
To graph the solution set, we need to plot the numbers greater than 4.5 on the number line. Since the inequality is strict (less than), we will use an open circle on 4.5 and shade to the right.
```
-----o----------->
4.5
```
In interval notation, this can be written as (4.5, ∞). The parentheses indicate that 4.5 is not included in the solution set, and the infinity symbol (∞) represents all numbers greater than 4.5.
Therefore, the solution to the inequality is (4.5, ∞) in interval notation.