Answer :
Let's translate the statement "16 less than a number [tex]\(x\)[/tex] is less than or equal to 0 or greater than 7" into a compound inequality.
1. Understanding "16 less than a number [tex]\(x\)[/tex]"
- This phrase can be expressed as [tex]\(x - 16\)[/tex].
2. "Is less than or equal to 0"
- We can write this part of the statement as the inequality [tex]\(x - 16 \leq 0\)[/tex].
3. "Is greater than 7"
- This part translates to the inequality [tex]\(x - 16 > 7\)[/tex].
4. Combining the inequalities with "or"
- The statement says "less than or equal to 0 or greater than 7," so we need to combine our two inequalities using "or."
5. Solving the inequalities
- For [tex]\(x - 16 \leq 0\)[/tex]: We add 16 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x - 16 \leq 0 \implies x \leq 16
\][/tex]
- For [tex]\(x - 16 > 7\)[/tex]: Similarly, we add 16 to both sides:
[tex]\[
x - 16 > 7 \implies x > 23
\][/tex]
6. Writing the compound inequality
- Combine the two inequalities with "or":
[tex]\[
x \leq 16 \quad \text{or} \quad x > 23
\][/tex]
This means the translated statement into a compound inequality is [tex]\(x \leq 16\)[/tex] or [tex]\(x > 23\)[/tex].
1. Understanding "16 less than a number [tex]\(x\)[/tex]"
- This phrase can be expressed as [tex]\(x - 16\)[/tex].
2. "Is less than or equal to 0"
- We can write this part of the statement as the inequality [tex]\(x - 16 \leq 0\)[/tex].
3. "Is greater than 7"
- This part translates to the inequality [tex]\(x - 16 > 7\)[/tex].
4. Combining the inequalities with "or"
- The statement says "less than or equal to 0 or greater than 7," so we need to combine our two inequalities using "or."
5. Solving the inequalities
- For [tex]\(x - 16 \leq 0\)[/tex]: We add 16 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x - 16 \leq 0 \implies x \leq 16
\][/tex]
- For [tex]\(x - 16 > 7\)[/tex]: Similarly, we add 16 to both sides:
[tex]\[
x - 16 > 7 \implies x > 23
\][/tex]
6. Writing the compound inequality
- Combine the two inequalities with "or":
[tex]\[
x \leq 16 \quad \text{or} \quad x > 23
\][/tex]
This means the translated statement into a compound inequality is [tex]\(x \leq 16\)[/tex] or [tex]\(x > 23\)[/tex].
