Of course, let's solve each inequality step-by-step.
### a) [tex]\(19 + x \leq 43\)[/tex]
1. Subtract 19 from both sides of the inequality to isolate [tex]\(x\)[/tex]:
[tex]\[
19 + x - 19 \leq 43 - 19
\][/tex]
2. Simplify the equation:
[tex]\[
x \leq 24
\][/tex]
So, [tex]\(x \leq 24\)[/tex].
### b) [tex]\(28 - x \geq 11\)[/tex]
1. Subtract 28 from both sides of the inequality:
[tex]\[
28 - x - 28 \geq 11 - 28
\][/tex]
2. Simplify the equation:
[tex]\[
-x \geq -17
\][/tex]
3. Multiply both sides by -1 (remember to flip the inequality sign when multiplying by a negative number):
[tex]\[
x \leq 17
\][/tex]
So, [tex]\(x \leq 17\)[/tex].
### c) [tex]\(x + 38 - 18 < 83\)[/tex]
1. Simplify the left side of the inequality first:
[tex]\[
x + 20 < 83
\][/tex]
2. Subtract 20 from both sides of the inequality:
[tex]\[
x + 20 - 20 < 83 - 20
\][/tex]
3. Simplify the equation:
[tex]\[
x < 63
\][/tex]
So, [tex]\(x < 63\)[/tex].
### d) [tex]\(41 + x - 39 < 83 - 77\)[/tex]
1. Simplify both sides of the inequality:
[tex]\[
(41 - 39) + x < (83 - 77)
\][/tex]
[tex]\[
2 + x < 6
\][/tex]
2. Subtract 2 from both sides of the inequality to isolate [tex]\(x\)[/tex]:
[tex]\[
2 + x - 2 < 6 - 2
\][/tex]
3. Simplify the equation:
[tex]\[
x < 4
\][/tex]
So, [tex]\(x < 4\)[/tex].
### Summary of Results
a) [tex]\(x \leq 24\)[/tex]
b) [tex]\(x \leq 17\)[/tex]
c) [tex]\(x < 63\)[/tex]
d) [tex]\(x < 4\)[/tex]
