Let’s start by solving each inequality step by step, and then we can discuss how each solution would look on a graph.
1. Solve the inequality:
\(-3(x - 12) > 0\)
Step 1: Distribute -3.
\(-3x + 36 > 0\)
Step 2: Solve for \(x\).
\(-3x > -36\)
Divide by -3 (remember to flip the inequality sign):
\(x < 12\)
2. Solve the inequality:
\(\frac{1}{4}(8x + 16) > -4\)
Step 1: Multiply both sides by 4 to eliminate the fraction.
\(8x + 16 > -16\)
Step 2: Subtract 16 from both sides.
\(8x > -32\)
Step 3: Divide by 8.
\(x > -4\)
3. Solve the inequality:
\(-5(x + 5) > -15\)
Step 1: Distribute -5.
\(-5x - 25 > -15\)
Step 2: Add 25 to both sides.
\(-5x > 10\)
Step 3: Divide by -5 (flip the inequality sign).
\(x < -2\)
4. Solve the inequality:
\(8x < 4x + 28\)
Step 1: Subtract \(4x\) from both sides.
\(4x < 28\)
Step 2: Divide by 4.
\(x < 7\)
Summary of Solutions:
- \(x < 12\)
- \(x > -4\)
- \(x < -2\)
- \(x < 7\)
Graphing the Solutions
Now, let's graph these inequalities on a number line:
- For \(x < 12\): A number line with an open circle at 12, shaded to the left.
- For \(x > -4\): A number line with an open circle at -4, shaded to the right.
- For \(x < -2\): A number line with an open circle at -2, shaded to the left.
- For \(x < 7\): A number line with an open circle at 7, shaded to the left.
Below is a basic representation of how the graphs would look if you were using a number line:
<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->
... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
(----------------------) <------------) <---)
(----------) <---)
Explanation of Graphs:
- For \(x < 12\), the line continues left from 12.
- For \(x > -4\), the line starts moving right from -4.
- For \(x < -2\), the line goes left from -2.
- For \(x < 7\), the line goes left from 7.
This is a rough graphical representation of the inequalities that depicts their solutions on the number line. Each one will have open circles at their respective points to denote that those points are not included in the solution.