At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Sure! Let's solve the given equation step by step.
The equation we have is:
[tex]\[ |3x + 5| + |2x - 3| = 25 \][/tex]
To solve this absolute value equation, we must consider different cases based on the points where the expressions inside the absolute values change sign. The critical points are where [tex]\(3x + 5 = 0\)[/tex] and [tex]\(2x - 3 = 0\)[/tex].
1. [tex]\(3x + 5 = 0\)[/tex]:
[tex]\[ x = -\frac{5}{3} \][/tex]
2. [tex]\(2x - 3 = 0\)[/tex]:
[tex]\[ x = \frac{3}{2} \][/tex]
We need to consider three intervals determined by these points:
- [tex]\(x < -\frac{5}{3}\)[/tex]
- [tex]\(-\frac{5}{3} \leq x \leq \frac{3}{2}\)[/tex]
- [tex]\(x > \frac{3}{2}\)[/tex]
### Case 1: [tex]\( x < -\frac{5}{3} \)[/tex]
Here, both [tex]\(3x + 5 < 0\)[/tex] and [tex]\(2x - 3 < 0\)[/tex]. Therefore, the absolute value expressions become:
[tex]\[ |3x + 5| = -(3x + 5) = -3x - 5 \][/tex]
[tex]\[ |2x - 3| = -(2x - 3) = -2x + 3 \][/tex]
So the equation becomes:
[tex]\[ -3x - 5 - 2x + 3 = 25 \][/tex]
[tex]\[ -5x - 2 = 25 \][/tex]
[tex]\[ -5x = 27 \][/tex]
[tex]\[ x = -\frac{27}{5} \][/tex]
We need to check if [tex]\( x = -\frac{27}{5} \)[/tex] falls in the interval [tex]\( x < -\frac{5}{3} \)[/tex]. Since [tex]\(-\frac{27}{5} = -5.4\)[/tex] and it is less than [tex]\(-\frac{5}{3} = -1.67\)[/tex], this value is valid.
### Case 2: [tex]\( -\frac{5}{3} \leq x \leq \frac{3}{2} \)[/tex]
Here, [tex]\(3x + 5 \geq 0\)[/tex] and [tex]\(2x - 3 < 0\)[/tex] when [tex]\(x < \frac{3}{2}\)[/tex]. Therefore, the absolute value expressions become:
[tex]\[ |3x + 5| = 3x + 5 \][/tex]
[tex]\[ |2x - 3| = -(2x - 3) = -2x + 3 \][/tex]
So the equation becomes:
[tex]\[ 3x + 5 - 2x + 3 = 25 \][/tex]
[tex]\[ x + 8 = 25 \][/tex]
[tex]\[ x = 17 \][/tex]
We need to check if [tex]\( x = 17 \)[/tex] falls in the interval [tex]\( -\frac{5}{3} \leq x \leq \frac{3}{2} \)[/tex]. Since [tex]\(17\)[/tex] is not in this interval, this value is not valid.
### Case 3: [tex]\( x > \frac{3}{2} \)[/tex]
Here, both [tex]\(3x + 5 \geq 0\)[/tex] and [tex]\(2x - 3 \geq 0\)[/tex]. Therefore, the absolute value expressions are:
[tex]\[ |3x + 5| = 3x + 5 \][/tex]
[tex]\[ |2x - 3| = 2x - 3 \][/tex]
So the equation becomes:
[tex]\[ 3x + 5 + 2x - 3 = 25 \][/tex]
[tex]\[ 5x + 2 = 25 \][/tex]
[tex]\[ 5x = 23 \][/tex]
[tex]\[ x = \frac{23}{5} \][/tex]
We need to check if [tex]\( x = \frac{23}{5} \)[/tex] falls in the interval [tex]\( x > \frac{3}{2} \)[/tex]. Since [tex]\(\frac{23}{5} = 4.6\)[/tex] and it is greater than [tex]\( \frac{3}{2} = 1.5 \)[/tex], this value is valid.
### Summary of Solutions
The valid solutions are:
- [tex]\(x = -\frac{27}{5}\)[/tex]
- [tex]\(x = \frac{23}{5}\)[/tex]
### Sum of Highest and Lowest Values
The highest value is [tex]\( \frac{23}{5} \)[/tex].
The lowest value is [tex]\(-\frac{27}{5}\)[/tex].
Their sum is:
[tex]\[ \frac{23}{5} + \left(-\frac{27}{5}\right) = \frac{23 - 27}{5} = \frac{-4}{5} = -0.8 \][/tex]
Therefore, the sum of the highest and lowest values is:
[tex]\[ \boxed{-0.8} \][/tex]
The equation we have is:
[tex]\[ |3x + 5| + |2x - 3| = 25 \][/tex]
To solve this absolute value equation, we must consider different cases based on the points where the expressions inside the absolute values change sign. The critical points are where [tex]\(3x + 5 = 0\)[/tex] and [tex]\(2x - 3 = 0\)[/tex].
1. [tex]\(3x + 5 = 0\)[/tex]:
[tex]\[ x = -\frac{5}{3} \][/tex]
2. [tex]\(2x - 3 = 0\)[/tex]:
[tex]\[ x = \frac{3}{2} \][/tex]
We need to consider three intervals determined by these points:
- [tex]\(x < -\frac{5}{3}\)[/tex]
- [tex]\(-\frac{5}{3} \leq x \leq \frac{3}{2}\)[/tex]
- [tex]\(x > \frac{3}{2}\)[/tex]
### Case 1: [tex]\( x < -\frac{5}{3} \)[/tex]
Here, both [tex]\(3x + 5 < 0\)[/tex] and [tex]\(2x - 3 < 0\)[/tex]. Therefore, the absolute value expressions become:
[tex]\[ |3x + 5| = -(3x + 5) = -3x - 5 \][/tex]
[tex]\[ |2x - 3| = -(2x - 3) = -2x + 3 \][/tex]
So the equation becomes:
[tex]\[ -3x - 5 - 2x + 3 = 25 \][/tex]
[tex]\[ -5x - 2 = 25 \][/tex]
[tex]\[ -5x = 27 \][/tex]
[tex]\[ x = -\frac{27}{5} \][/tex]
We need to check if [tex]\( x = -\frac{27}{5} \)[/tex] falls in the interval [tex]\( x < -\frac{5}{3} \)[/tex]. Since [tex]\(-\frac{27}{5} = -5.4\)[/tex] and it is less than [tex]\(-\frac{5}{3} = -1.67\)[/tex], this value is valid.
### Case 2: [tex]\( -\frac{5}{3} \leq x \leq \frac{3}{2} \)[/tex]
Here, [tex]\(3x + 5 \geq 0\)[/tex] and [tex]\(2x - 3 < 0\)[/tex] when [tex]\(x < \frac{3}{2}\)[/tex]. Therefore, the absolute value expressions become:
[tex]\[ |3x + 5| = 3x + 5 \][/tex]
[tex]\[ |2x - 3| = -(2x - 3) = -2x + 3 \][/tex]
So the equation becomes:
[tex]\[ 3x + 5 - 2x + 3 = 25 \][/tex]
[tex]\[ x + 8 = 25 \][/tex]
[tex]\[ x = 17 \][/tex]
We need to check if [tex]\( x = 17 \)[/tex] falls in the interval [tex]\( -\frac{5}{3} \leq x \leq \frac{3}{2} \)[/tex]. Since [tex]\(17\)[/tex] is not in this interval, this value is not valid.
### Case 3: [tex]\( x > \frac{3}{2} \)[/tex]
Here, both [tex]\(3x + 5 \geq 0\)[/tex] and [tex]\(2x - 3 \geq 0\)[/tex]. Therefore, the absolute value expressions are:
[tex]\[ |3x + 5| = 3x + 5 \][/tex]
[tex]\[ |2x - 3| = 2x - 3 \][/tex]
So the equation becomes:
[tex]\[ 3x + 5 + 2x - 3 = 25 \][/tex]
[tex]\[ 5x + 2 = 25 \][/tex]
[tex]\[ 5x = 23 \][/tex]
[tex]\[ x = \frac{23}{5} \][/tex]
We need to check if [tex]\( x = \frac{23}{5} \)[/tex] falls in the interval [tex]\( x > \frac{3}{2} \)[/tex]. Since [tex]\(\frac{23}{5} = 4.6\)[/tex] and it is greater than [tex]\( \frac{3}{2} = 1.5 \)[/tex], this value is valid.
### Summary of Solutions
The valid solutions are:
- [tex]\(x = -\frac{27}{5}\)[/tex]
- [tex]\(x = \frac{23}{5}\)[/tex]
### Sum of Highest and Lowest Values
The highest value is [tex]\( \frac{23}{5} \)[/tex].
The lowest value is [tex]\(-\frac{27}{5}\)[/tex].
Their sum is:
[tex]\[ \frac{23}{5} + \left(-\frac{27}{5}\right) = \frac{23 - 27}{5} = \frac{-4}{5} = -0.8 \][/tex]
Therefore, the sum of the highest and lowest values is:
[tex]\[ \boxed{-0.8} \][/tex]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
