Solve each compound inequality. Graph the solution. Show your work.
10≥7+3x and 9−4x≤1
First, let's solve the compound inequality 10 ≥ 7 + 3x.
Step 1: Subtract 7 from both sides of the inequality.
10 - 7 ≥ 7 + 3x - 7 simplify
3 ≥ 3x
Step 2: Divide both sides of the inequality by 3.
3/3 ≥ 3x/3 simplify
1 ≥ x
So the solution to the first compound inequality is x ≤ 1.
Next, let's solve the compound inequality 9 - 4x ≤ 1.
Step 1: Subtract 9 from both sides of the inequality.
9 - 9 - 4x ≤ 1 - 9 simplify
-4x ≤ -8
Step 2: Divide both sides of the inequality by -4. Note that when dividing by a negative number, the direction of the inequality sign must be reversed.
-4x/-4 ≥ -8/-4 simplify
x ≥ 2
So the solution to the second compound inequality is x ≥ 2.
Now let's graph the solutions on a number line.
For x ≤ 1, we plot a closed dot on the number 1 and shade the area to the left of the dot.
For x ≥ 2, we plot a closed dot on the number 2 and shade the area to the right of the dot.
The combined graph would look like this:
```
------(=)-------------------(=)-------
1 2
```
The shaded region represents the solution to the compound inequalities.