Question

match the following
-5(1 - 4x) > -5
-2(x + 1)+4 < 10
2x + 7 ≥ 27 or 3 + 3x ≤ 30
-1 < 9 + x < 17

Explanation:

1. Solve the inequality \(-5(1 - 4x)>- 5\):

• Step 1: Expand the left - hand side
• Using the distributive property \(a(b - c)=ab - ac\), we have \(-5\times1-(-5)\times4x>-5\), which simplifies to \(-5 + 20x>-5\).
• Step 2: Add 5 to both sides
• \(-5+5 + 20x>-5 + 5\), resulting in \(20x>0\).
• Step 3: Divide both sides by 20
• \(\frac{20x}{20}>\frac{0}{20}\), so \(x > 0\). The graph of \(x>0\) is a number line with an open - circle at 0 and a ray pointing to the right.

1. Solve the inequality \(-2(x + 1)+4<10\):

• Step 1: Expand the left - hand side
• Using the distributive property \(-2(x + 1)=-2x-2\), the inequality becomes \(-2x-2 + 4<10\), which simplifies to \(-2x+2<10\).
• Step 2: Subtract 2 from both sides
• \(-2x+2 - 2<10 - 2\), giving \(-2x<8\).
• Step 3: Divide both sides by \(-2\) and reverse the inequality sign
• \(\frac{-2x}{-2}>\frac{8}{-2}\) (since dividing by a negative number reverses the inequality), so \(x>-4\). The graph of \(x > - 4\) is a number line with an open - circle at \(-4\) and a ray pointing to the right.

1. Solve the compound inequality \(2x+7\geq27\) or \(3 + 3x\leq30\):

• Solve \(2x+7\geq27\):
• Step 1: Subtract 7 from both sides
• \(2x+7 - 7\geq27 - 7\), resulting in \(2x\geq20\).
• Step 2: Divide both sides by 2
• \(\frac{2x}{2}\geq\frac{20}{2}\), so \(x\geq10\).
• Solve \(3 + 3x\leq30\):
• Step 1: Subtract 3 from both sides
• \(3+3x - 3\leq30 - 3\), giving \(3x\leq27\).
• Step 2: Divide both sides by 3
• \(\frac{3x}{3}\leq\frac{27}{3}\), so \(x\leq9\). The graph of \(x\geq10\) or \(x\leq9\) has a closed - circle at 9, a closed - circle at 10, and rays pointing in opposite directions.

1. Solve the compound inequality \(-1<9 + x<17\):

• Step 1: Subtract 9 from all parts of the inequality
• \(-1-9<9 + x-9<17 - 9\), which simplifies to \(-10

Answer:

• \(-5(1 - 4x)>-5\) matches the number line with an open - circle at 0 and a ray pointing to the right.
• \(-2(x + 1)+4<10\) matches the number line with an open - circle at \(-4\) and a ray pointing to the right.
• \(2x+7\geq27\) or \(3 + 3x\leq30\) matches the number line with closed - circles at 9 and 10 and rays pointing in opposite directions.
• \(-1<9 + x<17\) matches the number line with open - circles at \(-10\) and 8 and a line segment between them.