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Answer :
To solve the inequality [tex]\(100 - 3x \geq -50\)[/tex], follow these steps:
1. Solve the related equation:
Start by solving the equation [tex]\(100 - 3x = -50\)[/tex] to find the boundary point.
[tex]\[
100 - 3x = -50
\][/tex]
Subtract 100 from both sides:
[tex]\[
-3x = -50 - 100
\][/tex]
[tex]\[
-3x = -150
\][/tex]
Divide both sides by -3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 50
\][/tex]
So, [tex]\(x = 50\)[/tex] is the point where the expression equals [tex]\(-50\)[/tex].
2. Test values less than, equal to, and greater than 50:
- [tex]\(x = 50\)[/tex]:
[tex]\[
100 - 3 \times 50 = 100 - 150 = -50
\][/tex]
This satisfies [tex]\(100 - 3x \geq -50\)[/tex] because [tex]\(-50 = -50\)[/tex].
- [tex]\(x > 50\)[/tex], e.g., [tex]\(x = 60\)[/tex]:
[tex]\[
100 - 3 \times 60 = 100 - 180 = -80
\][/tex]
-80 is less than -50, so it does not satisfy [tex]\(100 - 3x \geq -50\)[/tex].
- [tex]\(x < 50\)[/tex], e.g., [tex]\(x = 0\)[/tex]:
[tex]\[
100 - 3 \times 0 = 100 - 0 = 100
\][/tex]
100 is greater than -50, so it satisfies [tex]\(100 - 3x \geq -50\)[/tex].
- [tex]\(x < 50\)[/tex], e.g., [tex]\(x = -1\)[/tex]:
[tex]\[
100 - 3 \times (-1) = 100 + 3 = 103
\][/tex]
103 is also greater than -50, satisfying [tex]\(100 - 3x \geq -50\)[/tex].
3. Determine the solution to the inequality:
From these tests, we see that any [tex]\(x\)[/tex] value less than or equal to 50 satisfies the inequality [tex]\(100 - 3x \geq -50\)[/tex].
Hence, the solution to the inequality is [tex]\(x \leq 50\)[/tex].
1. Solve the related equation:
Start by solving the equation [tex]\(100 - 3x = -50\)[/tex] to find the boundary point.
[tex]\[
100 - 3x = -50
\][/tex]
Subtract 100 from both sides:
[tex]\[
-3x = -50 - 100
\][/tex]
[tex]\[
-3x = -150
\][/tex]
Divide both sides by -3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 50
\][/tex]
So, [tex]\(x = 50\)[/tex] is the point where the expression equals [tex]\(-50\)[/tex].
2. Test values less than, equal to, and greater than 50:
- [tex]\(x = 50\)[/tex]:
[tex]\[
100 - 3 \times 50 = 100 - 150 = -50
\][/tex]
This satisfies [tex]\(100 - 3x \geq -50\)[/tex] because [tex]\(-50 = -50\)[/tex].
- [tex]\(x > 50\)[/tex], e.g., [tex]\(x = 60\)[/tex]:
[tex]\[
100 - 3 \times 60 = 100 - 180 = -80
\][/tex]
-80 is less than -50, so it does not satisfy [tex]\(100 - 3x \geq -50\)[/tex].
- [tex]\(x < 50\)[/tex], e.g., [tex]\(x = 0\)[/tex]:
[tex]\[
100 - 3 \times 0 = 100 - 0 = 100
\][/tex]
100 is greater than -50, so it satisfies [tex]\(100 - 3x \geq -50\)[/tex].
- [tex]\(x < 50\)[/tex], e.g., [tex]\(x = -1\)[/tex]:
[tex]\[
100 - 3 \times (-1) = 100 + 3 = 103
\][/tex]
103 is also greater than -50, satisfying [tex]\(100 - 3x \geq -50\)[/tex].
3. Determine the solution to the inequality:
From these tests, we see that any [tex]\(x\)[/tex] value less than or equal to 50 satisfies the inequality [tex]\(100 - 3x \geq -50\)[/tex].
Hence, the solution to the inequality is [tex]\(x \leq 50\)[/tex].
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