Question

e) $2x \geq x - 1$
g) $\frac{x}{4} + 1 \geq 3$

1. represent on the real number line the values of $x$ such that

a) $3x + 10 \geq x + 6$
b) $5x - 1 < 3x + 1$
c) $3x - 2 \leq 4x + 1$
d) $-2x + 1 > 3x + 6$
e) $2(x - 1) - 3(x + 1) \leq 0$
f) $-2(x - 3) \geq 3(x - 1) - 1$

1. represent on the real number line the values of $x$ such that

a) $3 \leq 2x + 1 \leq 7$
b) $-4 < -3x + 2 < -1$
activity 7 solving an inequality in a given domain
given an inequality with one variable, the domain is the set of all possible values for the
a) consider the inequality $x + 1 \leq 4$. represent, on a number line, the solution of the inequality when the domain is:

1. $\mathbb{r}$
2. $\mathbb{z}$
3. $\mathbb{n}$

Explanation:

Let's solve the inequality \( x + 1 \leq 4 \) for different domains.

Part 1: Domain is \( \mathbb{R} \) (Real Numbers)

Step 1: Solve the inequality

Subtract 1 from both sides of the inequality \( x + 1 \leq 4 \):
\( x + 1 - 1 \leq 4 - 1 \)
\( x \leq 3 \)

Step 2: Represent on the number line

• Draw a real number line.
• Locate the point 3.
• Since \( x \leq 3 \), we draw a closed circle at 3 (because the inequality is "less than or equal to") and shade the line to the left of 3 (towards negative infinity).

Step 1: Solve the inequality (same as before)

From \( x + 1 \leq 4 \), we get \( x \leq 3 \).

Step 2: Find integer solutions

The integers \( x \) that satisfy \( x \leq 3 \) are \( \dots, -2, -1, 0, 1, 2, 3 \).

Step 3: Represent on the number line (for integers)

• Draw a number line with integer markings.
• Mark closed circles (or dots) at each integer value from negative infinity up to and including 3.

Step 1: Solve the inequality (same as before)

From \( x + 1 \leq 4 \), we get \( x \leq 3 \).

Step 2: Find natural number solutions

The natural numbers \( x \) that satisfy \( x \leq 3 \) are \( 0, 1, 2, 3 \) (if \( \mathbb{N} \) includes 0) or \( 1, 2, 3 \) (if \( \mathbb{N} \) starts at 1). Commonly, in many curricula, \( \mathbb{N} \) starts at 0 or 1, but let's check the inequality:

If \( \mathbb{N} = \{0, 1, 2, 3, \dots\} \), then \( x = 0, 1, 2, 3 \) satisfy \( x \leq 3 \).

If \( \mathbb{N} = \{1, 2, 3, \dots\} \), then \( x = 1, 2, 3 \) satisfy \( x \leq 3 \).

Step 3: Represent on the number line (for natural numbers)

• Draw a number line with natural number markings (0, 1, 2, 3, ... or 1, 2, 3, ... depending on definition).
• Mark closed circles (or dots) at the natural numbers \( x \) where \( x \leq 3 \).

Answer:

(for representation on number line description):
The solution \( x \leq 3 \) on the real number line is represented by a closed circle at 3 and a line segment (or ray) extending to the left from 3.

Part 2: Domain is \( \mathbb{Z} \) (Integers)