Answer :
Sure! Let's solve the inequality step-by-step to find the solution set for [tex]\(2(2 - 4x) + 3(2x + 6) < 8\)[/tex].
1. Expand the expressions:
- Distribute the 2 in the first term: [tex]\(2 \times (2 - 4x) = 4 - 8x\)[/tex]
- Distribute the 3 in the second term: [tex]\(3 \times (2x + 6) = 6x + 18\)[/tex]
2. Write the expanded inequality:
[tex]\[
4 - 8x + 6x + 18 < 8
\][/tex]
3. Combine like terms:
- Combine the constant terms: [tex]\(4 + 18 = 22\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-8x + 6x = -2x\)[/tex]
So, the inequality becomes:
[tex]\[
22 - 2x < 8
\][/tex]
4. Isolate the [tex]\(x\)[/tex] term:
- Subtract 22 from both sides:
[tex]\[
-2x < 8 - 22
\][/tex]
[tex]\[
-2x < -14
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
- Divide both sides by -2, and remember to reverse the inequality sign when you divide by a negative number:
[tex]\[
x > 7
\][/tex]
Therefore, the solution set for the inequality is [tex]\(x > 7\)[/tex].
The correct answer is c. [tex]\(x > 7\)[/tex].
1. Expand the expressions:
- Distribute the 2 in the first term: [tex]\(2 \times (2 - 4x) = 4 - 8x\)[/tex]
- Distribute the 3 in the second term: [tex]\(3 \times (2x + 6) = 6x + 18\)[/tex]
2. Write the expanded inequality:
[tex]\[
4 - 8x + 6x + 18 < 8
\][/tex]
3. Combine like terms:
- Combine the constant terms: [tex]\(4 + 18 = 22\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-8x + 6x = -2x\)[/tex]
So, the inequality becomes:
[tex]\[
22 - 2x < 8
\][/tex]
4. Isolate the [tex]\(x\)[/tex] term:
- Subtract 22 from both sides:
[tex]\[
-2x < 8 - 22
\][/tex]
[tex]\[
-2x < -14
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
- Divide both sides by -2, and remember to reverse the inequality sign when you divide by a negative number:
[tex]\[
x > 7
\][/tex]
Therefore, the solution set for the inequality is [tex]\(x > 7\)[/tex].
The correct answer is c. [tex]\(x > 7\)[/tex].
