Question

solve each inequality and graph its solution.

1. $|b + 9| > 13$
2. $|a + 4| \geq 5$

Explanation:

Problem 23: Solve \(|b + 9|>13\) and graph the solution

Step 1: Apply absolute value inequality rule

For \(|x|>k\) (where \(k>0\)), the solution is \(x < -k\) or \(x>k\). Here, \(x = b + 9\) and \(k = 13\), so we get two inequalities:
\(b + 9 < - 13\) or \(b + 9>13\)

Step 2: Solve the first inequality \(b + 9 < - 13\)

Subtract 9 from both sides:
\(b+9 - 9<-13 - 9\)
\(b < - 22\)

Step 3: Solve the second inequality \(b + 9>13\)

Subtract 9 from both sides:
\(b + 9-9>13 - 9\)
\(b>4\)

Step 4: Graph the solution

On the number line, we have an open circle at \(-22\) (since \(b < - 22\), not including \(-22\)) and an arrow to the left, and an open circle at \(4\) (since \(b>4\), not including \(4\)) and an arrow to the right. The number line provided has marks at \(-24,-20,-16,\cdots,4\). So we mark an open circle at \(-22\) (between \(-24\) and \(-20\)) and draw an arrow left, and an open circle at \(4\) and draw an arrow right.

Step 1: Apply absolute value inequality rule

For \(|x|\geq k\) (where \(k > 0\)), the solution is \(x\leq - k\) or \(x\geq k\). Here, \(x=a + 4\) and \(k = 5\), so we get two inequalities:
\(a + 4\leq - 5\) or \(a + 4\geq5\)

Step 2: Solve the first inequality \(a + 4\leq - 5\)

Subtract 4 from both sides:
\(a+4 - 4\leq - 5-4\)
\(a\leq - 9\)

Step 3: Solve the second inequality \(a + 4\geq5\)

Subtract 4 from both sides:
\(a + 4-4\geq5 - 4\)
\(a\geq1\)

Step 4: Graph the solution

On the number line, we have a closed circle at \(-9\) (since \(a\leq - 9\), including \(-9\)) and an arrow to the left, and a closed circle at \(1\) (since \(a\geq1\), including \(1\)) and an arrow to the right. The number line provided has marks at \(-14,-12,\cdots,4\). So we mark a closed circle at \(-9\) (between \(-12\) and \(-8\)) and draw an arrow left, and a closed circle at \(1\) (between \(0\) and \(2\)) and draw an arrow right.

Answer:

(for inequality): \(b < - 22\) or \(b>4\)

Problem 24: Solve \(|a + 4|\geq5\) and graph the solution