Question

Solve: \(x-2 \leq-4 x+3 \leq x+18\)

Short answer

The solution is \(-3 \leq x \leq 1\).

Step-by-step solution

- Split the Compound Inequality

The given compound inequality is: \[x-2 \leq -4x + 3 \leq x + 18\]This can be split into two separate inequalities: 1. \(x - 2 \leq -4x + 3\)2. \(-4x + 3 \leq x + 18\)

- Solve the First Inequality

Solve \(x - 2 \leq -4x + 3\):Add \(4x\) to both sides:\[x + 4x - 2 \leq -4x + 4x + 3\]Simplify:\[5x - 2 \leq 3\]Add 2 to both sides:\[5x - 2 + 2 \leq 3 + 2\]Simplify:\[5x \leq 5\]Divide both sides by 5:\[x \leq 1\]

- Solve the Second Inequality

Solve \(-4x + 3 \leq x + 18\):Add \(4x\) to both sides:\[-4x + 4x + 3 \leq x + 4x + 18\]Simplify:\[3 \leq 5x + 18\]Subtract 18 from both sides:\[3 - 18 \leq 5x + 18 - 18\]Simplify:\[-15 \leq 5x\]Divide both sides by 5:\[\frac{-15}{5} \leq \frac{5x}{5}\]Simplify:\[-3 \leq x\]

- Combine the Inequalities

Combine the results from Step 2 and Step 3:\[-3 \leq x \leq 1\]

Key concepts

Solving Inequalities

When solving inequalities, the goal is to find the range of values that satisfy the inequality. The process is similar to solving equations, but with an important consideration: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
In the given exercise, we started with a compound inequality, which is essentially two inequalities combined into one statement. Here’s a brief overview:

• We split the compound inequality into two simpler inequalities.
• We solved each of these inequalities separately.
• Finally, we combined the solutions to get the final answer.

This systematic approach ensures that we cover all possible values that satisfy the given inequality.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying terms to isolate the variable. In our exercise, we used several key steps:
For the first inequality, \( x - 2 \leq -4x + 3 \):

• Add 4x to both sides to move all x terms to one side.
• Simplify and then isolate x by performing basic arithmetic operations like addition and division.

Here's the detailed breakdown:
\[ x - 2 \leq -4x + 3 \]Add 4x to both sides: \[ x + 4x - 2 \leq -4x + 4x + 3 \] Simplifies to: \[ 5x - 2 \leq 3 \] Add 2 to both sides: \[ 5x - 2 + 2 \leq 3 + 2 \] Simplifies to: \[ 5x \leq 5 \] Divide both sides by 5: \[ x \leq 1 \]
This same process is applied to the second inequality. Mastering these manipulative steps makes solving any type of linear inequalities straightforward.

Combining Inequalities

Combining inequalities is the final step in solving compound inequalities. After solving the individual inequalities, you combine the results to identify the range of values that satisfies the entire compound inequality.
For our exercise, the results from the two separate inequalities \[ x - 2 \leq -4x + 3 \] and \[ -4x + 3 \leq x + 18 \] were \[ x \leq 1 \] and \[ -3 \leq x \] respectively.
Combining these, we get:
\[ -3 \leq x \leq 1 \]
This combined inequality tells us that x can be any value between -3 and 1, inclusive.
Remember, when you combine inequalities, you're looking for the intersection of their solution sets. This gives us the overall solution to the compound inequality, ensuring all parts are satisfied.