Save the inequality, and plot the solftion on real number
Ine

Ask by Gardner Lawson. in the United Arab Emirates

Mar 20,2025

Question

Save the inequality, and plot the solftion on real number Ine $$ 3 x+5 \leq 5-3(x+2) \leq 6 x-10 \quad \forall x \in R $$

UpStudy AI · Accepted Answer

We are given the chain inequality
$$
3x+5 \leq 5-3(x+2) \leq 6x-10.
$$
We will break this into two separate inequalities and solve each.

**Step 1. Solve the left inequality**

$$
3x+5 \leq 5-3(x+2).
$$

First, expand the right side:
$$
5-3(x+2)=5-3x-6=-3x-1.
$$

So, the inequality becomes:
$$
3x+5 \leq -3x-1.
$$

Now, add $$3x$$ to both sides:
$$
6x+5 \leq -1.
$$

Subtract 5 from both sides:
$$
6x \leq -6.
$$

Divide by 6:
$$
x \leq -1.
$$

**Step 2. Solve the right inequality**

$$
5-3(x+2) \leq 6x-10.
$$

Again, expand the left side:
$$
5-3(x+2)=5-3x-6=-3x-1.
$$

Thus, the inequality becomes:
$$
-3x-1 \leq 6x-10.
$$

Add $$3x$$ to both sides:
$$
-1 \leq 9x-10.
$$

Add 10 to both sides:
$$
9 \leq 9x.
$$

Divide by 9:
$$
x \geq 1.
$$

**Step 3. Combine the results**

The system requires:
$$
x \leq -1 \quad \text{and} \quad x \geq 1.
$$

No real number satisfies both conditions simultaneously. Therefore, the solution set is the empty set:
$$
\varnothing.
$$

**Step 4. Plot on the real number line**

Since there is no solution, the plot on the real number line is empty (i.e., there is no $$x$$-value that satisfies the inequality).

Thus, the final answer is: $$\varnothing$$.
There is no solution to the inequality, so the answer is the empty set: ∅.

Save the inequality, and plot the solftion on real number
Ine