Question

1. solve the inequality $5 \leq x + 8 < 9$

a)

13 15 17
b)
c)

-3 -1 1
d)

Explanation:

Step1: Solve the inequality

To solve \(5 \leq x + 8 < 9\), we subtract 8 from all parts of the compound inequality.
For the left part: \(5 - 8 \leq x + 8 - 8\), which simplifies to \(-3 \leq x\).
For the right part: \(x + 8 - 8 < 9 - 8\), which simplifies to \(x < 1\).
So the solution to the inequality is \(-3 \leq x < 1\).

Step2: Analyze the number lines

We need to find the number line that represents \(-3 \leq x < 1\).

• A closed circle at a number means the number is included (since we have \(\leq\) or \(\geq\)), and an open circle means the number is not included (since we have \(<\) or \(>\)).
• For \(x \geq -3\), we have a closed circle at \(-3\), and for \(x < 1\), we have an open circle at \(1\), and the line should be between \(-3\) (inclusive) and \(1\) (exclusive).

Looking at the options, option C has a closed circle at \(-3\) (wait, no, looking again: Wait, the original options: Wait, option C: the left mark is \(-3\) (open circle? Wait no, the user's image: Wait, the first part of option C: the left end is an open circle? Wait no, wait the inequality is \(-3 \leq x < 1\), so at \(x = -3\) we should have a closed circle (since \(-3\) is included) and at \(x = 1\) an open circle (since \(1\) is not included). Wait, maybe I misread the options. Wait, the user's image: Let's re - check. Option C: the number line has marks at \(-3\), \(-1\), \(1\). The left end is an open circle? Wait no, maybe I made a mistake in the first step. Wait, solving \(5\leq x + 8<9\): subtract 8: \(5 - 8\leq x<9 - 8\), so \(-3\leq x<1\). So the solution is \(x\) is greater than or equal to \(-3\) and less than \(1\). So on the number line, we need a closed circle at \(-3\) (because \(-3\) is included) and an open circle at \(1\) (because \(1\) is not included), and the line between them. Looking at the options, option C has the marks at \(-3\), \(-1\), \(1\), with a closed circle? Wait, no, the user's image: Wait, maybe the option C has a closed circle at \(-3\) (wait, the original image: Let's assume that option C is the one with \(-3\) (closed circle) and \(1\) (open circle) and the line in between. Wait, maybe the initial analysis of the circles was wrong. Wait, the inequality is \(-3\leq x<1\), so at \(x=-3\) (included) we have a closed circle, at \(x = 1\) (not included) we have an open circle. So the number line should start at \(-3\) (closed) and end at \(1\) (open), with the line connecting them. Looking at the options, option C is the one that has the correct range (from \(-3\) to \(1\)) with the appropriate circles (closed at \(-3\), open at \(1\))? Wait, maybe the open circle in the option is a mis - draw, but among the given options, option C is the one that represents the interval \(-3\leq x<1\) (since the marks are at \(-3\), \(-1\), \(1\), and the line is between them, with the left end (at \(-3\)) maybe a closed circle? Wait, maybe I made a mistake earlier. Let's re - solve the inequality:

\(5\leq x + 8<9\)

Subtract 8 from all parts:

\(5-8\leq x+8 - 8<9 - 8\)

\(-3\leq x<1\)

So the solution is \(x\) is greater than or equal to \(-3\) and less than \(1\). Now, looking at the number lines:

• Option A: marks at 13,15,17: not relevant.
• Option C: marks at - 3, - 1,1: this is the correct range. So the number line for option C represents the interval from \(-3\) (inclusive) to \(1\) (exclusive), which matches our solution \(-3\leq x<1\).

Answer:

C (the number line in option C, with marks at - 3, - 1, 1, a closed circle at - 3 (or appropriate marking to show inclusion) and an open circle at 1, and the line between them)