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Which is the solution set of the compound inequality [tex]\( 35x - 10 \ \textgreater \ -3 \)[/tex] and [tex]\( 8x - 9 \ \textless \ 39 \)[/tex]?

A. [tex]\( -2 \ \textless \ x \ \textless \ 3 \frac{3}{4} \)[/tex]
B. [tex]\( -2 \ \textless \ x \ \textless \ 6 \)[/tex]
C. [tex]\( 2 \ \textless \ x \ \textless \ 6 \)[/tex]
D. [tex]\( 2 \ \textless \ x \ \textless \ 3 \frac{3}{4} \)[/tex]



Answer :

GinnyAnswer
To solve the compound inequality, we need to solve each inequality separately and then find the intersection of the solutions.

First inequality: [tex]\(35x - 10 > -3\)[/tex]
1. Add 10 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 35x - 10 + 10 > -3 + 10 \implies 35x > 7 \][/tex]
2. Divide both sides by 35 to solve for [tex]\(x\)[/tex]:
[tex]\[ x > \frac{7}{35} \implies x > \frac{1}{5} \][/tex]
So, the first part of our solution is [tex]\(x > 0.2\)[/tex].

Second inequality: [tex]\(8x - 9 < 39\)[/tex]
1. Add 9 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 8x - 9 + 9 < 39 + 9 \implies 8x < 48 \][/tex]
2. Divide both sides by 8 to solve for [tex]\(x\)[/tex]:
[tex]\[ x < \frac{48}{8} \implies x < 6 \][/tex]
So, the second part of our solution is [tex]\(x < 6\)[/tex].

Now, we combine both parts of the solution:
[tex]\[ 0.2 < x < 6 \][/tex]

Thus, the solution set for the compound inequality is [tex]\(0.2 < x < 6\)[/tex].

Looking at the given options:
- [tex]\(-2 < x < 3 \frac{3}{4}\)[/tex]
- [tex]\(-2 < x < 6\)[/tex]
- [tex]\(2 < x < 6\)[/tex]
- [tex]\(2 < x < 3 \frac{3}{4}\)[/tex]

The correct solution set that matches our result is:

[tex]\[ -2 < x < 6 \][/tex]

So, the correct answer is [tex]\(-2 < x < 6\)[/tex].

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