Question

for each inequality, show the integers which are solutions in two ways: by making a list and by graphing.
-5 -4 -3 -2 -1 0 1 2 3 4 5
list
graph
x < 1 { }
x > 5 { }
x ≤ -3 { }
x ≥ -4 { }
x > 210 { }
x ≤ 0 { }
|x| < 4 { }
|x| ≥ 2 { }
x + 1 ≥ 5 { }
©1990 by key curriculum press. do not duplicate without permission. 15

Explanation:

Step1: Analyze the inequality type

For strict inequalities (\(<\) or \(>\)), use open - circles on the number line. For non - strict inequalities (\(\leq\) or \(\geq\)), use closed - circles.

Step2: Determine integer solutions

For \(x < 1\), integers less than \(1\) are found. For \(x>5\), integers greater than \(5\) are found, and so on.

Step3: Rewrite absolute - value inequalities

For \(|x|<4\), it means \(-4 < x < 4\) and for \(|x|\geq2\), it means \(x\leq - 2\) or \(x\geq2\). Then find integer solutions.

Step4: Solve linear inequality

For \(x + 1\geq5\), solve for \(x\) by subtracting \(1\) from both sides to get \(x\geq4\), then find integer solutions.

Answer:

1. For \(x < 1\):

• List: \(\{...,-3,-2,-1,0\}\)
• Graph: An open - circle at \(1\) and an arrow pointing to the left on the number line.

1. For \(x>5\):

• List: \(\{6,7,8,...\}\)
• Graph: An open - circle at \(5\) and an arrow pointing to the right on the number line.

1. For \(x\leq - 3\):

• List: \(\{...,-5,-4,-3\}\)
• Graph: A closed - circle at \(-3\) and an arrow pointing to the left on the number line.

1. For \(x\geq - 4\):

• List: \(\{-4,-3,-2,...\}\)
• Graph: A closed - circle at \(-4\) and an arrow pointing to the right on the number line.

1. For \(x > 210\):

• List: \(\{211,212,213,...\}\)
• Graph: An open - circle at \(210\) and an arrow pointing to the right on the number line.

1. For \(x\leq0\):

• List: \(\{...,-2,-1,0\}\)
• Graph: A closed - circle at \(0\) and an arrow pointing to the left on the number line.

1. For \(|x|<4\):

• First, rewrite as \(-4 < x < 4\).
• List: \(\{-3,-2,-1,0,1,2,3\}\)
• Graph: Open - circles at \(-4\) and \(4\) and an arrow between them on the number line.

1. For \(|x|\geq2\):

• Rewrite as \(x\leq - 2\) or \(x\geq2\).
• List: \(\{...,-4,-3,-2,2,3,4,...\}\)
• Graph: Closed - circles at \(-2\) and \(2\) with arrows pointing away from each other (left from \(-2\) and right from \(2\)) on the number line.

1. For \(x + 1\geq5\), solve for \(x\):

• Subtract \(1\) from both sides: \(x\geq4\).
• List: \(\{4,5,6,...\}\)
• Graph: A closed - circle at \(4\) and an arrow pointing to the right on the number line.