Solve for xx and graph the solution on the number line below.

Question

minus, x, minus, 5, is greater than or equal to, 1, or, minus, x, minus, 5, is less than, minus, 8
−x−5≥1or
−x−5<−8
Answer
Attempt 1 out of 2


Inequality Notation:
Number Line:

Answer 1

To solve the inequalities given, let’s break them down step by step.

First Inequality:

\[ -x - 5 \geq 1 \]

1. Add 5 to both sides: \[ -x \geq 6 \]
2. Multiply by -1 (remember to reverse the inequality): \[ x \leq -6 \]

Second Inequality:

\[ -x - 5 < -8 \]

1. Add 5 to both sides: \[ -x < -3 \]
2. Multiply by -1 (remember to reverse the inequality): \[ x > 3 \]

Solution in Inequality Notation:

The solutions to the inequalities can be combined with "or": \[ x \leq -6 \quad \text{or} \quad x > 3 \]

Number Line Representation:

• To represent \(x \leq -6\), we draw a solid circle on -6 and shade to the left.
• To represent \(x > 3\), we draw an open circle on 3 and shade to the right.

Summary:

1. Inequality Notation: \[ x \leq -6 \quad \text{or} \quad x > 3 \]

2. Number Line:

<---|---|---|---|---|---|---|---|---|---|---|---|---> 
    -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4 
       ●---------------------------  
                       ○         
           x > 3

• Here, there is a closed circle at -6, shaded to the left, indicating \(x \leq -6\), and an open circle at 3, shaded to the right, indicating \(x > 3\).

Answer 2

Solve for xx and graph the solution on the number line below.
minus, 11, is less than or equal to, minus, 3, x, minus, 8, or, minus, 17, is greater than or equal to, minus, 3, x, minus, 8
−11≤−3x−8or
−17≥−3x−8
Answer
Attempt 1 out of 2


Inequality Notation:
Number Line:

Answer 3

To solve the inequalities given, we'll break them down step by step.

First Inequality:

\[ -11 \leq -3x - 8 \]

1. Add 8 to both sides: \[ -11 + 8 \leq -3x \] \[ -3 \leq -3x \]
2. Divide by -3 (remember to reverse the inequality): \[ 1 \geq x \quad \text{or} \quad x \leq 1 \]

Second Inequality:

\[ -17 \geq -3x - 8 \]

1. Add 8 to both sides: \[ -17 + 8 \geq -3x \] \[ -9 \geq -3x \]
2. Divide by -3 (remember to reverse the inequality): \[ 3 \leq x \quad \text{or} \quad x \geq 3 \]

Solution in Inequality Notation:

The solutions to the inequalities can be combined with "or": \[ x \leq 1 \quad \text{or} \quad x \geq 3 \]

Number Line Representation:

• To represent \(x \leq 1\), we draw a solid circle on 1 and shade to the left.
• To represent \(x \geq 3\), we draw a solid circle on 3 and shade to the right.

Summary:

1. Inequality Notation: \[ x \leq 1 \quad \text{or} \quad x \geq 3 \]

2. Number Line:

<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---> 
    -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4 
       ●-------------------------  
                               ●------------------- 
                                   x ≥ 3

• Here, there is a closed circle at 1, shaded to the left, indicating \(x \leq 1\), and another closed circle at 3, shaded to the right, indicating \(x \geq 3\).

Answer 4

Solve for xx and graph the solution on the number line below.
minus, x, minus, 3, is greater than or equal to, minus, 2, or, minus, x, minus, 3, is less than or equal to, minus, 9
−x−3≥−2or
−x−3≤−9
Answer
Attempt 1 out of 2


Inequality Notation:
Number Line:

Answer 5

To solve the inequalities given, we need to break them down into two parts and solve each one step by step.

First Inequality:

\[ -x - 3 \geq -2 \]

1. Add 3 to both sides: \[ -x \geq 1 \]
2. Multiply by -1 (remember to reverse the inequality): \[ x \leq -1 \]

Second Inequality:

\[ -x - 3 \leq -9 \]

1. Add 3 to both sides: \[ -x \leq -6 \]
2. Multiply by -1 (remember to reverse the inequality): \[ x \geq 6 \]

Combine Solutions:

The solutions to the inequalities can be combined with "or": \[ x \leq -1 \quad \text{or} \quad x \geq 6 \]

Summary:

1. Inequality Notation: \[ x \leq -1 \quad \text{or} \quad x \geq 6 \]

2. Number Line Representation:

• To represent \(x \leq -1\), we draw a solid circle at -1 and shade to the left.
• To represent \(x \geq 6\), we draw a solid circle at 6 and shade to the right.

Number Line:

<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---> 
    -3  -2  -1   0   1   2   3   4   5   6   7   8   9
       ●-------------------   
                          ●--------------------

• Here, there is a closed circle at -1, shaded to the left, indicating \(x \leq -1\), and a closed circle at 6, shaded to the right, indicating \(x \geq 6\).