Question

match the following absolute value inequalities with their corresponding solutions.
|m - 2| < 8
|-8x - 3| > 11
|x| - 3 > 2
1 + |1 + x| < 4
9|x - 8| - 10 < 26
|x| + 5 ≥ 11
(number lines and no solution option are shown as part of the image)

Explanation:

To solve and match each absolute - value inequality with its solution, we analyze each inequality one by one:

1. Solve \(|m - 2|\lt8\)

The absolute - value inequality \(|A|\lt B\) (where \(B\gt0\)) is equivalent to \(-B\lt A\lt B\).
For \(|m - 2|\lt8\), we have \(-8\lt m - 2\lt8\).
Add 2 to all parts of the compound inequality:
\(-8 + 2\lt m-2 + 2\lt8 + 2\)
\(-6\lt m\lt10\)
The number line for this solution has open circles at \(-6\) and \(10\) and the line segment between them.

2. Solve \(|-8x - 3|\gt11\)

The absolute - value inequality \(|A|\gt B\) (where \(B\gt0\)) is equivalent to \(A\gt B\) or \(A\lt - B\).
For \(|-8x - 3|\gt11\), we have two cases:

• Case 1: \(-8x-3\gt11\)

Add 3 to both sides: \(-8x\gt11 + 3=-8x\gt14\)
Divide both sides by \(-8\) (and reverse the inequality sign): \(x\lt-\frac{14}{8}=-\frac{7}{4}=-1.75\)

• Case 2: \(-8x - 3\lt - 11\)

Add 3 to both sides: \(-8x\lt-11 + 3=-8x\lt - 8\)
Divide both sides by \(-8\) (and reverse the inequality sign): \(x\gt1\)
The solution is \(x\lt - 1.75\) or \(x\gt1\), which is represented by two rays going in opposite directions (open circles at the boundary points).

3. Solve \(|x|-3\gt2\)

First, isolate the absolute - value term: \(|x|\gt2 + 3=|x|\gt5\)
The absolute - value inequality \(|x|\gt5\) is equivalent to \(x\gt5\) or \(x\lt - 5\)
The number line for this solution has two rays, one to the right of \(5\) and one to the left of \(-5\) (open circles at \(5\) and \(-5\)).

4. Solve \(1+|1 + x|\lt4\)

First, isolate the absolute - value term: \(|1 + x|\lt4 - 1=|1 + x|\lt3\)
The inequality \(|A|\lt B\) (where \(B\gt0\)) is equivalent to \(-B\lt A\lt B\). So, \(-3\lt1 + x\lt3\)
Subtract 1 from all parts: \(-3-1\lt1 + x-1\lt3 - 1\)
\(-4\lt x\lt2\)
The number line has open circles at \(-4\) and \(2\) and the line segment between them.

5. Solve \(9|x - 8|-10\lt26\)

First, isolate the absolute - value term:
\(9|x - 8|\lt26 + 10=9|x - 8|\lt36\)
Divide both sides by 9: \(|x - 8|\lt4\)
The inequality \(|A|\lt B\) (where \(B\gt0\)) is equivalent to \(-B\lt A\lt B\). So, \(-4\lt x - 8\lt4\)
Add 8 to all parts: \(-4 + 8\lt x-8 + 8\lt4 + 8\)
\(4\lt x\lt12\)
The number line has open circles at \(4\) and \(12\) and the line segment between them.

6. Solve \(|x|+5\geq11\)

First, isolate the absolute - value term: \(|x|\geq11 - 5=|x|\geq6\)
The absolute - value inequality \(|x|\geq6\) is equivalent to \(x\geq6\) or \(x\leq - 6\)
The number line has two rays, one to the right of \(6\) (including \(6\)) and one to the left of \(-6\) (including \(-6\)) (closed circles at \(6\) and \(-6\)).

Now, we can match each inequality with its corresponding number - line solution:

• \(|m - 2|\lt8\): Solution is \(-6\lt m\lt10\) (matches the number line with open circles at \(-6\) and \(10\) and the segment between them).
• \(|-8x - 3|\gt11\): Solution is \(x\lt - 1.75\) or \(x\gt1\) (matches the number line with two rays, one left of \(-2\) (approx) and one right of \(1\)).
• \(|x|-3\gt2\): Solution is \(x\gt5\) or \(x\lt - 5\) (matches the number line with two rays, one left of \(-5\) and one right of \(5\)).
• \(1+|1 + x|\lt4\): Solution is \(-4\lt x\lt2\) (matches the number line with open circles at \(-4\) and \(2\) and the segment between them).
• \(9|x - 8|-10\lt26\): Solution is \(4\lt x\lt12\) (matches the number line with open circles at \(4\) and \(12\) and the segment between them).
• \(|x|+5\geq11\): Solution is \(x\geq6\) or \(x\leq - 6\) (matches the number line with closed circles at \(-6\) and \(6\) and two rays).

Answer:

To solve and match each absolute - value inequality with its solution, we analyze each inequality one by one:

1. Solve \(|m - 2|\lt8\)

The absolute - value inequality \(|A|\lt B\) (where \(B\gt0\)) is equivalent to \(-B\lt A\lt B\).
For \(|m - 2|\lt8\), we have \(-8\lt m - 2\lt8\).
Add 2 to all parts of the compound inequality:
\(-8 + 2\lt m-2 + 2\lt8 + 2\)
\(-6\lt m\lt10\)
The number line for this solution has open circles at \(-6\) and \(10\) and the line segment between them.

2. Solve \(|-8x - 3|\gt11\)

The absolute - value inequality \(|A|\gt B\) (where \(B\gt0\)) is equivalent to \(A\gt B\) or \(A\lt - B\).
For \(|-8x - 3|\gt11\), we have two cases:

• Case 1: \(-8x-3\gt11\)

Add 3 to both sides: \(-8x\gt11 + 3=-8x\gt14\)
Divide both sides by \(-8\) (and reverse the inequality sign): \(x\lt-\frac{14}{8}=-\frac{7}{4}=-1.75\)

• Case 2: \(-8x - 3\lt - 11\)

Add 3 to both sides: \(-8x\lt-11 + 3=-8x\lt - 8\)
Divide both sides by \(-8\) (and reverse the inequality sign): \(x\gt1\)
The solution is \(x\lt - 1.75\) or \(x\gt1\), which is represented by two rays going in opposite directions (open circles at the boundary points).

3. Solve \(|x|-3\gt2\)

First, isolate the absolute - value term: \(|x|\gt2 + 3=|x|\gt5\)
The absolute - value inequality \(|x|\gt5\) is equivalent to \(x\gt5\) or \(x\lt - 5\)
The number line for this solution has two rays, one to the right of \(5\) and one to the left of \(-5\) (open circles at \(5\) and \(-5\)).

4. Solve \(1+|1 + x|\lt4\)

First, isolate the absolute - value term: \(|1 + x|\lt4 - 1=|1 + x|\lt3\)
The inequality \(|A|\lt B\) (where \(B\gt0\)) is equivalent to \(-B\lt A\lt B\). So, \(-3\lt1 + x\lt3\)
Subtract 1 from all parts: \(-3-1\lt1 + x-1\lt3 - 1\)
\(-4\lt x\lt2\)
The number line has open circles at \(-4\) and \(2\) and the line segment between them.

5. Solve \(9|x - 8|-10\lt26\)

First, isolate the absolute - value term:
\(9|x - 8|\lt26 + 10=9|x - 8|\lt36\)
Divide both sides by 9: \(|x - 8|\lt4\)
The inequality \(|A|\lt B\) (where \(B\gt0\)) is equivalent to \(-B\lt A\lt B\). So, \(-4\lt x - 8\lt4\)
Add 8 to all parts: \(-4 + 8\lt x-8 + 8\lt4 + 8\)
\(4\lt x\lt12\)
The number line has open circles at \(4\) and \(12\) and the line segment between them.

6. Solve \(|x|+5\geq11\)

First, isolate the absolute - value term: \(|x|\geq11 - 5=|x|\geq6\)
The absolute - value inequality \(|x|\geq6\) is equivalent to \(x\geq6\) or \(x\leq - 6\)
The number line has two rays, one to the right of \(6\) (including \(6\)) and one to the left of \(-6\) (including \(-6\)) (closed circles at \(6\) and \(-6\)).

Now, we can match each inequality with its corresponding number - line solution:

• \(|m - 2|\lt8\): Solution is \(-6\lt m\lt10\) (matches the number line with open circles at \(-6\) and \(10\) and the segment between them).
• \(|-8x - 3|\gt11\): Solution is \(x\lt - 1.75\) or \(x\gt1\) (matches the number line with two rays, one left of \(-2\) (approx) and one right of \(1\)).
• \(|x|-3\gt2\): Solution is \(x\gt5\) or \(x\lt - 5\) (matches the number line with two rays, one left of \(-5\) and one right of \(5\)).
• \(1+|1 + x|\lt4\): Solution is \(-4\lt x\lt2\) (matches the number line with open circles at \(-4\) and \(2\) and the segment between them).
• \(9|x - 8|-10\lt26\): Solution is \(4\lt x\lt12\) (matches the number line with open circles at \(4\) and \(12\) and the segment between them).
• \(|x|+5\geq11\): Solution is \(x\geq6\) or \(x\leq - 6\) (matches the number line with closed circles at \(-6\) and \(6\) and two rays).