a) Rewrite 2x 5 3 without absolute value sign. [3] b) Find the domain and the range of the following relation R = \(1,1), (2,4), (3,9), (4,16)\ [3]

a) Rewriting $|2x-5|\le 3$ without absolute value sign

The inequality $|2x-5|\le 3$ involves an absolute value. To rewrite it without the absolute value, we use the definition of absolute value inequalities. The general form $|A|\le B$ (where $B>0$) can be rewritten as $-B\le A\le B$.

Applying this to $|2x-5|\le 3$:

$$-3\le 2x-5\le 3$$

Now, we solve the compound inequality step-by-step:

1. Add 5 to all parts of the inequality to isolate the term with $x$: $$-3+5\le 2x-5+5\le 3+5$$ Simplifying: $$2\le 2x\le 8$$

2. Divide all parts by 2 to solve for $x$: $$\frac{2}{2}\le \frac{2x}{2}\le \frac{8}{2}$$ Simplifying: $$1\le x\le 4$$

Thus, the inequality $|2x-5|\le 3$ can be rewritten without the absolute value sign as:

$$\boxed{1\le x\le 4}$$

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b) Finding the domain and range of the relation $R=\{(1,1),(2,4),(3,9),(4,16)\}$

Domain

The domain of a relation is the set of all first elements (input values) of the ordered pairs in the relation.

Given the relation: $$R=\{(1,1),(2,4),(3,9),(4,16)\}$$

The first elements of the ordered pairs are: 1, 2, 3, 4.

Thus, the domain of $R$ is: $$\boxed{\{1,2,3,4\}}$$

Range

The range of a relation is the set of all second elements (output values) of the ordered pairs in the relation.

The second elements of the ordered pairs are: 1, 4, 9, 16.

Thus, the range of $R$ is: $$\boxed{\{1,4,9,16\}}$$

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Additional Explanation (for deeper understanding)

Absolute Value Inequality

The inequality $|2x-5|\le 3$ can be interpreted as the distance between $2x-5$ and 0 on the number line is less than or equal to 3. This means $2x-5$ lies between $-3$ and $3$, which is why we rewrite it as a compound inequality.

Relation and Domain/Range

The relation $R$ is given as a set of ordered pairs. Each ordered pair $(a,b)$ means that $a$ is mapped to $b$. The domain is the set of all $a$ values, and the range is the set of all $b$ values.

Observing the Pattern

If we observe the relation $R$ more closely, we notice that the second elements (outputs) are perfect squares of the first elements (inputs): $$1=1^2,4=2^2,9=3^2,16=4^2$$ This suggests that the relation $R$ represents the function $y=x^2$ for $x=1,2,3,4$. However, since the domain is explicitly given as $\{1,2,3,4\}$, the range is the set of squares of these values.