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Answer :
Let's solve the inequalities step by step to determine the restrictions on [tex]\( x \)[/tex].
1. First Inequality:
[tex]\[
87 + 3x + 35 < 180
\][/tex]
- Combine the constants:
[tex]\[
87 + 35 = 122
\][/tex]
- Rewrite the inequality:
[tex]\[
122 + 3x < 180
\][/tex]
- Subtract 122 from both sides:
[tex]\[
3x < 180 - 122
\][/tex]
- Calculate:
[tex]\[
3x < 58
\][/tex]
- Divide by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x < \frac{58}{3}
\][/tex]
- Simplify:
[tex]\[
x < 19.33
\][/tex]
2. Second Inequality:
[tex]\[
8x + 120 < 180
\][/tex]
- Subtract 120 from both sides:
[tex]\[
8x < 180 - 120
\][/tex]
- Calculate:
[tex]\[
8x < 60
\][/tex]
- Divide by 8 to solve for [tex]\( x \)[/tex]:
[tex]\[
x < \frac{60}{8}
\][/tex]
- Simplify:
[tex]\[
x < 7.5
\][/tex]
3. Third Inequality:
[tex]\[
3x < 20
\][/tex]
- Divide by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x < \frac{20}{3}
\][/tex]
- Simplify:
[tex]\[
x < 6.67
\][/tex]
Conclusion:
To determine the overall restriction on [tex]\( x \)[/tex], we need the value that satisfies all three inequalities. Therefore, [tex]\( x \)[/tex] must be less than the smallest value from the solutions:
- [tex]\( x < 19.33 \)[/tex]
- [tex]\( x < 7.5 \)[/tex]
- [tex]\( x < 6.67 \)[/tex]
The most restrictive condition is [tex]\( x < 6.67 \)[/tex]. Thus, the restriction on [tex]\( x \)[/tex] is:
[tex]\[
x < 6.67
\][/tex]
1. First Inequality:
[tex]\[
87 + 3x + 35 < 180
\][/tex]
- Combine the constants:
[tex]\[
87 + 35 = 122
\][/tex]
- Rewrite the inequality:
[tex]\[
122 + 3x < 180
\][/tex]
- Subtract 122 from both sides:
[tex]\[
3x < 180 - 122
\][/tex]
- Calculate:
[tex]\[
3x < 58
\][/tex]
- Divide by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x < \frac{58}{3}
\][/tex]
- Simplify:
[tex]\[
x < 19.33
\][/tex]
2. Second Inequality:
[tex]\[
8x + 120 < 180
\][/tex]
- Subtract 120 from both sides:
[tex]\[
8x < 180 - 120
\][/tex]
- Calculate:
[tex]\[
8x < 60
\][/tex]
- Divide by 8 to solve for [tex]\( x \)[/tex]:
[tex]\[
x < \frac{60}{8}
\][/tex]
- Simplify:
[tex]\[
x < 7.5
\][/tex]
3. Third Inequality:
[tex]\[
3x < 20
\][/tex]
- Divide by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x < \frac{20}{3}
\][/tex]
- Simplify:
[tex]\[
x < 6.67
\][/tex]
Conclusion:
To determine the overall restriction on [tex]\( x \)[/tex], we need the value that satisfies all three inequalities. Therefore, [tex]\( x \)[/tex] must be less than the smallest value from the solutions:
- [tex]\( x < 19.33 \)[/tex]
- [tex]\( x < 7.5 \)[/tex]
- [tex]\( x < 6.67 \)[/tex]
The most restrictive condition is [tex]\( x < 6.67 \)[/tex]. Thus, the restriction on [tex]\( x \)[/tex] is:
[tex]\[
x < 6.67
\][/tex]
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