Answer :
To solve the inequality [tex]\( |3t - 2| \leq 10 \)[/tex], follow these steps:
### Step-by-Step Solution:
1. Understand the Meaning of Absolute Value:
The absolute value of a number represents its distance from zero on the number line, which is always non-negative. For an expression like [tex]\( |3t - 2| \leq 10 \)[/tex], this can be interpreted as:
[tex]\[ -10 \leq 3t - 2 \leq 10 \][/tex]
2. Remove the Absolute Value by Splitting the Inequality:
The given absolute value inequality can be broken down into two separate linear inequalities:
[tex]\[ -10 \leq 3t - 2 \quad \text{and} \quad 3t - 2 \leq 10 \][/tex]
3. Solve Each Inequality Separately:
- For the first inequality:
[tex]\[ -10 \leq 3t - 2 \][/tex]
Add 2 to both sides:
[tex]\[ -10 + 2 \leq 3t \][/tex]
[tex]\[ -8 \leq 3t \][/tex]
Divide by 3:
[tex]\[ -\frac{8}{3} \leq t \][/tex]
- For the second inequality:
[tex]\[ 3t - 2 \leq 10 \][/tex]
Add 2 to both sides:
[tex]\[ 3t \leq 10 + 2 \][/tex]
[tex]\[ 3t \leq 12 \][/tex]
Divide by 3:
[tex]\[ t \leq 4 \][/tex]
4. Combine the Solutions:
Both conditions must be satisfied simultaneously. Therefore, combining the inequalities:
[tex]\[ -\frac{8}{3} \leq t \leq 4 \][/tex]
### Interpret the Result:
The solution set for the inequality [tex]\( |3t - 2| \leq 10 \)[/tex] is the interval:
[tex]\[ -\frac{8}{3} \leq t \leq 4 \][/tex]
### Final Answer:
In interval notation, the solution set is:
[tex]\[ \boxed{[-\frac{8}{3}, 4]} \][/tex]
Since there are infinitely many values of [tex]\( t \)[/tex] between [tex]\( -\frac{8}{3} \)[/tex] and [tex]\( 4 \)[/tex], we select Option A.
### Graphical Representation:
The solution set can be represented on a number line with a closed interval:
- A solid dot at [tex]\( -\frac{8}{3} \)[/tex] and a solid dot at [tex]\( 4 \)[/tex], indicating that the endpoints are included.
- A solid line connecting these two points, representing all values [tex]\( t \)[/tex] between [tex]\( -\frac{8}{3} \)[/tex] and [tex]\( 4 \)[/tex].
### Conclusion:
The correct choice is:
A. There are infinitely many solutions. The solution set is [tex]\([- \frac{8}{3}, 4]\)[/tex].
### Step-by-Step Solution:
1. Understand the Meaning of Absolute Value:
The absolute value of a number represents its distance from zero on the number line, which is always non-negative. For an expression like [tex]\( |3t - 2| \leq 10 \)[/tex], this can be interpreted as:
[tex]\[ -10 \leq 3t - 2 \leq 10 \][/tex]
2. Remove the Absolute Value by Splitting the Inequality:
The given absolute value inequality can be broken down into two separate linear inequalities:
[tex]\[ -10 \leq 3t - 2 \quad \text{and} \quad 3t - 2 \leq 10 \][/tex]
3. Solve Each Inequality Separately:
- For the first inequality:
[tex]\[ -10 \leq 3t - 2 \][/tex]
Add 2 to both sides:
[tex]\[ -10 + 2 \leq 3t \][/tex]
[tex]\[ -8 \leq 3t \][/tex]
Divide by 3:
[tex]\[ -\frac{8}{3} \leq t \][/tex]
- For the second inequality:
[tex]\[ 3t - 2 \leq 10 \][/tex]
Add 2 to both sides:
[tex]\[ 3t \leq 10 + 2 \][/tex]
[tex]\[ 3t \leq 12 \][/tex]
Divide by 3:
[tex]\[ t \leq 4 \][/tex]
4. Combine the Solutions:
Both conditions must be satisfied simultaneously. Therefore, combining the inequalities:
[tex]\[ -\frac{8}{3} \leq t \leq 4 \][/tex]
### Interpret the Result:
The solution set for the inequality [tex]\( |3t - 2| \leq 10 \)[/tex] is the interval:
[tex]\[ -\frac{8}{3} \leq t \leq 4 \][/tex]
### Final Answer:
In interval notation, the solution set is:
[tex]\[ \boxed{[-\frac{8}{3}, 4]} \][/tex]
Since there are infinitely many values of [tex]\( t \)[/tex] between [tex]\( -\frac{8}{3} \)[/tex] and [tex]\( 4 \)[/tex], we select Option A.
### Graphical Representation:
The solution set can be represented on a number line with a closed interval:
- A solid dot at [tex]\( -\frac{8}{3} \)[/tex] and a solid dot at [tex]\( 4 \)[/tex], indicating that the endpoints are included.
- A solid line connecting these two points, representing all values [tex]\( t \)[/tex] between [tex]\( -\frac{8}{3} \)[/tex] and [tex]\( 4 \)[/tex].
### Conclusion:
The correct choice is:
A. There are infinitely many solutions. The solution set is [tex]\([- \frac{8}{3}, 4]\)[/tex].
