Question

directions: solve each compound or double - inequality and then match the problem up with the graph of its solution.

1. solve: (x + 2>4) and (x - 1<4)
2. solve: (0leq x + 3leq5)
3. solve: (2 + xleq2) or (2-3xgeq - 4)
4. solve: (4xleq - 12) and (\frac{x}{2}leq1)
5. solve: (\frac{x}{2}+3leq1) or (\frac{5x}{4}-1>\frac{3}{2})
6. solve: (2x - 5<5) or (3x + 1>-2)

Explanation:

Step1: Solve the first compound - inequality

For \(x + 2>4\) and \(x - 1<4\):
Solve \(x+2>4\):
Subtract 2 from both sides: \(x+2 - 2>4 - 2\), so \(x>2\).
Solve \(x - 1<4\):
Add 1 to both sides: \(x-1 + 1<4 + 1\), so \(x<5\). The solution is \(2

Step2: Solve the second compound - inequality

For \(0\leq x + 3\leq5\):
Subtract 3 from all parts: \(0-3\leq x+3 - 3\leq5 - 3\), so \(-3\leq x\leq2\).

Step3: Solve the third compound - inequality

For \(2 + x\leq2\) or \(2-3x\geq - 4\):
Solve \(2 + x\leq2\), subtract 2 from both sides: \(x\leq0\).
Solve \(2-3x\geq - 4\), subtract 2 from both sides: \(2-3x-2\geq - 4 - 2\), \(-3x\geq - 6\). Divide both sides by - 3 and reverse the inequality sign, so \(x\leq2\). The solution of the 'or' - inequality is \(x\leq2\).

Step4: Solve the fourth compound - inequality

For \(4x\leq - 12\) and \(\frac{x}{2}\leq1\):
Solve \(4x\leq - 12\), divide both sides by 4: \(x\leq - 3\).
Solve \(\frac{x}{2}\leq1\), multiply both sides by 2: \(x\leq2\). The solution is \(x\leq - 3\).

Step5: Solve the fifth compound - inequality

For \(\frac{x}{2}+3\leq1\) or \(\frac{5x}{4}-1>\frac{3}{2}\):
Solve \(\frac{x}{2}+3\leq1\), subtract 3 from both sides: \(\frac{x}{2}\leq - 2\), multiply both sides by 2: \(x\leq - 4\).
Solve \(\frac{5x}{4}-1>\frac{3}{2}\), add 1 to both sides: \(\frac{5x}{4}>\frac{3 + 2}{2}=\frac{5}{2}\), multiply both sides by \(\frac{4}{5}\): \(x>2\).

Step6: Solve the sixth compound - inequality

For \(2x-5<5\) or \(3x + 1>-2\):
Solve \(2x-5<5\), add 5 to both sides: \(2x<10\), divide both sides by 2: \(x<5\).
Solve \(3x + 1>-2\), subtract 1 from both sides: \(3x>-3\), divide both sides by 3: \(x>-1\). The solution of the 'or' - inequality is all real numbers.

1. The solution \(2
2. The solution \(-3\leq x\leq2\) matches the graph with closed - circles at - 3 and 2 and the line between them.
3. The solution \(x\leq2\) matches the graph with a closed - circle at 2 and the line going to the left.
4. The solution \(x\leq - 3\) matches the graph with a closed - circle at - 3 and the line going to the left.
5. The solution \(x\leq - 4\) or \(x>2\) matches the graph with a closed - circle at - 4 and an open - circle at 2 and lines going in opposite directions.
6. The solution of all real numbers matches the graph with arrows going in both directions.

Answer:

1. Match with the graph that has open - circles at 2 and 5 and a line segment between them.
2. Match with the graph that has closed - circles at - 3 and 2 and a line segment between them.
3. Match with the graph that has a closed - circle at 2 and a line going to the left.
4. Match with the graph that has a closed - circle at - 3 and a line going to the left.
5. Match with the graph that has a closed - circle at - 4, an open - circle at 2 and lines going in opposite directions.
6. Match with the graph that has arrows going in both directions.