1 a. consider the inequality -1 ≤ $\frac{x}{2}$. i. predict which values of x will make the inequality true. ii. complete the table to check your prediction. (table with x values -4, -3, -2, -1, 0, 1, 2, 3, 4 and $\frac{x}{2}$ column) b. consider the inequality 1 ≤ $\frac{x}{2}$. i. predict which values of x will make it true. ii. complete the table to check your prediction. (table with x values -4, -3, -2, -1, 0, 1, 2, 3, 4 and $\frac{x}{2}$ column) 2 diego is solving the inequality 100 - 3x ≥ -50. first, he solves the equation 100 - 3x = -50 and gets x = 50. which inequality represents all the solutions for 100 - 3x ≥ -50? a x 50 d x ≥ 50 3 solve the inequality -5(x - 1) -40, and graph the solution on the number line. (number line diagram)

Question

1 a. consider the inequality -1 ≤ $\frac{x}{2}$. i. predict which values of x will make the inequality true. ii. complete the table to check your prediction. (table with x values -4, -3, -2, -1, 0, 1, 2, 3, 4 and $\frac{x}{2}$ column) b. consider the inequality 1 ≤ $\frac{x}{2}$. i. predict which values of x will make it true. ii. complete the table to check your prediction. (table with x values -4, -3, -2, -1, 0, 1, 2, 3, 4 and $\frac{x}{2}$ column) 2 diego is solving the inequality 100 - 3x ≥ -50. first, he solves the equation 100 - 3x = -50 and gets x = 50. which inequality represents all the solutions for 100 - 3x ≥ -50? a x < 50 b x ≤ 50 c x > 50 d x ≥ 50 3 solve the inequality -5(x - 1) > -40, and graph the solution on the number line. (number line diagram)

Accepted Answer

### Part 1a i
# Explanation:
## Step1: Analyze the inequality
We have the inequality $-1 \leq \frac{x}{2}$. To find the values of $x$ that satisfy this, we can multiply both sides by 2 (since 2 is positive, the inequality sign remains the same).
## Step2: Solve for $x$
Multiplying both sides by 2 gives: $-1\times2 \leq \frac{x}{2}\times2$, which simplifies to $-2 \leq x$ or $x \geq -2$. So we predict that all values of $x$ greater than or equal to -2 will make the inequality true.
# Answer:
All real numbers $x$ such that $x \geq -2$ (e.g., $x=-2, -1, 0, 1, 2, \dots$)

### Part 1a ii
# Explanation:
## Step1: Calculate $\frac{x}{2}$ for each $x$
For $x = -4$: $\frac{-4}{2}=-2$
For $x = -3$: $\frac{-3}{2}=-1.5$
For $x = -2$: $\frac{-2}{2}=-1$
For $x = -1$: $\frac{-1}{2}=-0.5$
For $x = 0$: $\frac{0}{2}=0$
For $x = 1$: $\frac{1}{2}=0.5$
For $x = 2$: $\frac{2}{2}=1$
For $x = 3$: $\frac{3}{2}=1.5$
For $x = 4$: $\frac{4}{2}=2$
## Step2: Fill the table
| $x$ | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|-------|----|----|----|----|---|---|---|---|---|
| $\frac{x}{2}$ | -2 | -1.5 | -1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 |
# Answer:
The table is filled as above.

### Part 1b i
# Explanation:
## Step1: Analyze the inequality
We have the inequality $1 \leq \frac{x}{2}$. Multiply both sides by 2 (positive, so inequality sign remains).
## Step2: Solve for $x$
Multiplying both sides by 2: $1\times2 \leq \frac{x}{2}\times2$, so $2 \leq x$ or $x \geq 2$. We predict $x$ values greater than or equal to 2 make it true.
# Answer:
All real numbers $x$ such that $x \geq 2$ (e.g., $x = 2, 3, 4, \dots$)

### Part 1b ii
# Explanation:
## Step1: Calculate $\frac{x}{2}$ for each $x$
For $x = -4$: $\frac{-4}{2}=-2$
For $x = -3$: $\frac{-3}{2}=-1.5$
For $x = -2$: $\frac{-2}{2}=-1$
For $x = -1$: $\frac{-1}{2}=-0.5$
For $x = 0$: $\frac{0}{2}=0$
For $x = 1$: $\frac{1}{2}=0.5$
For $x = 2$: $\frac{2}{2}=1$
For $x = 3$: $\frac{3}{2}=1.5$
For $x = 4$: $\frac{4}{2}=2$
## Step2: Fill the table
| $x$ | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|-------|----|----|----|----|---|---|---|---|---|
| $\frac{x}{2}$ | -2 | -1.5 | -1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 |
# Answer:
The table is filled as above.

### Question 2
# Explanation:
## Step1: Analyze the inequality $100 - 3x \geq -50$
First, we know the boundary solution from $100 - 3x=-50$ is $x = 50$. Now, let's test a value less than 50, say $x = 0$: $100-3(0)=100 \geq -50$ (true). Test a value greater than 50, say $x = 60$: $100 - 3(60)=100 - 180=-80 \geq -50$? No, -80 < -50. So the inequality holds when $x \leq 50$.
## Step2: Match with options
Option B is $x \leq 50$, which matches our analysis.
# Answer:
B. $x \leq 50$

### Question 3
# Explanation:
## Step1: Solve the inequality $-5(x - 1) > -40$
First, divide both sides by -5. Remember, when dividing by a negative number, the inequality sign flips. So $\frac{-5(x - 1)}{-5} < \frac{-40}{-5}$
## Step2: Simplify
Simplifying gives $x - 1 < 8$
## Step3: Solve for $x$
Add 1 to both sides: $x - 1 + 1 < 8 + 1$, so $x < 9$
## Step4: Graph on number line
On the number line, we draw an open circle at 9 (since $x < 9$, not including 9) and shade the region to the left of 9.
# Answer:
The solution to the inequality is $x < 9$. For the number line graph: Draw an open circle at 9 and shade the line to the left of 9.

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QUESTION IMAGE

Question

Explanation:

Part 1a i

Step1: Analyze the inequality

Step2: Solve for \(x\)

Step1: Calculate \(\frac{x}{2}\) for each \(x\)

Step2: Fill the table

Step1: Analyze the inequality

Step2: Solve for \(x\)

Answer:

Part 1a ii