Of course! Let's break down and solve each equation step-by-step:
### Problem 11:
Solve [tex]\(2x - 3 = 7\)[/tex].
1. Add 3 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
2x - 3 + 3 = 7 + 3
\][/tex]
[tex]\[
2x = 10
\][/tex]
2. Divide both sides by 2:
[tex]\[
x = \frac{10}{2}
\][/tex]
[tex]\[
x = 5
\][/tex]
### Problem 12:
Solve [tex]\(\frac{x}{3} + \frac{5}{2} = \frac{-3}{2}\)[/tex].
1. Multiply every term by the least common multiple of the denominators (which is 6) to clear the fractions:
[tex]\[
6 \left(\frac{x}{3}\right) + 6 \left(\frac{5}{2}\right) = 6 \left(\frac{-3}{2}\right)
\][/tex]
[tex]\[
2x + 15 = -9
\][/tex]
2. Subtract 15 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
2x + 15 - 15 = -9 - 15
\][/tex]
[tex]\[
2x = -24
\][/tex]
3. Divide both sides by 2:
[tex]\[
x = \frac{-24}{2}
\][/tex]
[tex]\[
x = -12
\][/tex]
### Problem 13:
Solve [tex]\(\frac{x + 6}{4} + \frac{x - 3}{5} = \frac{5x - 4}{8}\)[/tex].
1. Multiply every term by the least common multiple of the denominators (which is 40) to clear the fractions:
[tex]\[
40 \left(\frac{x + 6}{4}\right) + 40 \left(\frac{x - 3}{5}\right) = 40 \left(\frac{5x - 4}{8}\right)
\][/tex]
[tex]\[
10(x + 6) + 8(x - 3) = 5(5x - 4)
\][/tex]
2. Simplify and combine like terms:
[tex]\[
10x + 60 + 8x - 24 = 25x - 20
\][/tex]
[tex]\[
18x + 36 = 25x - 20
\][/tex]
3. Subtract 18x from both sides:
[tex]\[
18x + 36 - 18x = 25x - 20 - 18x
\][/tex]
[tex]\[
36 = 7x - 20
\][/tex]
4. Add 20 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
36 + 20 = 7x - 20 + 20
\][/tex]
[tex]\[
56 = 7x
\][/tex]
5. Divide both sides by 7:
[tex]\[
x = \frac{56}{7}
\][/tex]
[tex]\[
x = 8
\][/tex]
### Problem 14:
Solve [tex]\(\frac{6x + 1}{3} + 1 = \frac{x - 3}{6}\)[/tex].
1. Multiply every term by the least common multiple of the denominators (which is 6) to clear the fractions:
[tex]\[
6 \left(\frac{6x + 1}{3}\right) + 6 \cdot 1 = 6 \left(\frac{x - 3}{6}\right)
\][/tex]
[tex]\[
2(6x + 1) + 6 = x - 3
\][/tex]
[tex]\[
12x + 2 + 6 = x - 3
\][/tex]
2. Simplify and combine like terms:
[tex]\[
12x + 8 = x - 3
\][/tex]
3. Subtract [tex]\(x\)[/tex] from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
12x + 8 - x = x - 3 - x
\][/tex]
[tex]\[
11x + 8 = -3
\][/tex]
4. Subtract 8 from both sides:
[tex]\[
11x + 8 - 8 = -3 - 8
\][/tex]
[tex]\[
11x = -11
\][/tex]
5. Divide both sides by 11:
[tex]\[
x = \frac{-11}{11}
\][/tex]
[tex]\[
x = -1
\][/tex]
### Problem 15:
Solve [tex]\(2x + 5 = x + 25\)[/tex].
1. Subtract [tex]\(x\)[/tex] from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
2x + 5 - x = x + 25 - x
\][/tex]
[tex]\[
x + 5 = 25
\][/tex]
2. Subtract 5 from both sides:
[tex]\[
x + 5 - 5 = 25 - 5
\][/tex]
[tex]\[
x = 20
\][/tex]
So the solutions are:
1. [tex]\(x = 5\)[/tex]
2. [tex]\(x = -12\)[/tex]
3. [tex]\(x = 8\)[/tex]
4. [tex]\(x = -1\)[/tex]
5. [tex]\(x = 20\)[/tex]
