Answer :
Sure, let's solve the problem step-by-step and understand what the solution means.
We have two inequalities to solve:
1. [tex]\(-13 \leq -x - 7\)[/tex]
2. [tex]\(-16 \geq -x - 7\)[/tex]
Solving the first inequality:
[tex]\[ -13 \leq -x - 7 \][/tex]
- Add 7 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ -13 + 7 \leq -x \][/tex]
Simplifying this,
[tex]\[ -6 \leq -x \][/tex]
- Next, multiply both sides by -1 to solve for [tex]\(x\)[/tex], remembering that multiplying or dividing by a negative number flips the inequality sign:
[tex]\[ 6 \geq x \][/tex]
Or, equivalently,
[tex]\[ x \leq 6 \][/tex]
Solving the second inequality:
[tex]\[ -16 \geq -x - 7 \][/tex]
- Add 7 to both sides:
[tex]\[ -16 + 7 \geq -x \][/tex]
Simplifying,
[tex]\[ -9 \geq -x \][/tex]
- Again, multiply both sides by -1 (flip the inequality sign):
[tex]\[ 9 \leq x \][/tex]
Or, equivalently,
[tex]\[ x \geq 9 \][/tex]
Combining the solutions:
Now, we combine our results from both inequalities. We're looking at:
- [tex]\(x \leq 6\)[/tex]
- [tex]\(x \geq 9\)[/tex]
In inequality notation, this union of sets is expressed as:
[tex]\[ x \leq 6 \quad \text{or} \quad x \geq 9 \][/tex]
Graphing the solution on the number line:
1. Draw a number line.
2. Place a closed circle at 6 and shade to the left, indicating all numbers less than or equal to 6.
3. Place another closed circle at 9 and shade to the right, indicating all numbers greater than or equal to 9.
This graph represents the solution to the inequalities. The solution interval shows where the conditions for [tex]\(x\)[/tex] hold true.
If you have any further questions or need more clarification, feel free to ask!
We have two inequalities to solve:
1. [tex]\(-13 \leq -x - 7\)[/tex]
2. [tex]\(-16 \geq -x - 7\)[/tex]
Solving the first inequality:
[tex]\[ -13 \leq -x - 7 \][/tex]
- Add 7 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ -13 + 7 \leq -x \][/tex]
Simplifying this,
[tex]\[ -6 \leq -x \][/tex]
- Next, multiply both sides by -1 to solve for [tex]\(x\)[/tex], remembering that multiplying or dividing by a negative number flips the inequality sign:
[tex]\[ 6 \geq x \][/tex]
Or, equivalently,
[tex]\[ x \leq 6 \][/tex]
Solving the second inequality:
[tex]\[ -16 \geq -x - 7 \][/tex]
- Add 7 to both sides:
[tex]\[ -16 + 7 \geq -x \][/tex]
Simplifying,
[tex]\[ -9 \geq -x \][/tex]
- Again, multiply both sides by -1 (flip the inequality sign):
[tex]\[ 9 \leq x \][/tex]
Or, equivalently,
[tex]\[ x \geq 9 \][/tex]
Combining the solutions:
Now, we combine our results from both inequalities. We're looking at:
- [tex]\(x \leq 6\)[/tex]
- [tex]\(x \geq 9\)[/tex]
In inequality notation, this union of sets is expressed as:
[tex]\[ x \leq 6 \quad \text{or} \quad x \geq 9 \][/tex]
Graphing the solution on the number line:
1. Draw a number line.
2. Place a closed circle at 6 and shade to the left, indicating all numbers less than or equal to 6.
3. Place another closed circle at 9 and shade to the right, indicating all numbers greater than or equal to 9.
This graph represents the solution to the inequalities. The solution interval shows where the conditions for [tex]\(x\)[/tex] hold true.
If you have any further questions or need more clarification, feel free to ask!
