Solve the Linear Inequality: -2x + 9 ≤ 3 Step by Step

Q: Why do I have to flip the inequality sign when dividing by a negative?

Think about it: if $ 2 is true, then \( -2 > -4 $ must also be true! Multiplying or dividing by a negative number reverses the order of numbers on the number line.

Q: How can I remember when to flip the sign?

Create a simple rule: "Negative operation = flip the sign!" Whenever you multiply or divide both sides by a negative number, the inequality arrow must point the opposite direction.

Q: What's the difference between ≤ and < in my answer?

≤ means "less than or equal to" while Since our original inequality used ≤, we keep the "equal to" part in our final answer.

Q: How do I check if x ≥ 3 is correct?

Pick any number ≥ 3 (like x = 5) and substitute: $ -2(5) + 9 = -1 \leq 3 $ ✓. Also try the boundary: $ -2(3) + 9 = 3 \leq 3 $ ✓. Both work!

Q: Why did the explanation say the answer is x ≤ 3 at the end?

That's actually an error in the explanation! The correct answer is $ x \geq 3 $, as shown in all the work above. Always double-check the final statement matches your calculations.

Linear Inequalities with Sign Reversal

Solve the inequality:

$-2x + 9 \leq 3$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the inequality:

$-2x + 9 \leq 3$

Step-by-step solution

To solve the inequality $-2x + 9 \leq 3$ , follow these steps:

1. Subtract 9 from both sides:

$-2x + 9 - 9 \leq 3 - 9$

2. This simplifies to:

$-2x \leq -6$

3. Divide each side by -2, remembering to reverse the inequality sign since dividing by a negative number:

$x \geq \frac{-6}{-2}$

4. Simplifying gives:

$x \geq 3$

Thus, the solution is $x \leq 3$ .

Final Answer

$x \geq 3$

Key Points to Remember

Essential concepts to master this topic

Rule: Reverse inequality sign when dividing by negative number
Technique: -2x ≤ -6 becomes x ≥ 3 after dividing by -2
Check: Substitute x = 3: -2(3) + 9 = 3 ✓ and x = 4: -2(4) + 9 = 1 < 3 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to flip the inequality sign
Don't keep ≤ when dividing by -2 = wrong solution x ≤ 3! This completely reverses which values satisfy the inequality. Always flip the inequality sign whenever you multiply or divide both sides by a negative number.

Practice Quiz

Test your knowledge with interactive questions

Solve the following inequality:

$3x+4 \leq 10$

FAQ

Everything you need to know about this question

Why do I have to flip the inequality sign when dividing by a negative?

Think about it: if $2 < 4$ is true, then $-2 > -4$ must also be true! Multiplying or dividing by a negative number reverses the order of numbers on the number line.

How can I remember when to flip the sign?

Create a simple rule: "Negative operation = flip the sign!" Whenever you multiply or divide both sides by a negative number, the inequality arrow must point the opposite direction.

What's the difference between ≤ and < in my answer?

≤ means "less than or equal to" while < means "strictly less than." Since our original inequality used ≤, we keep the "equal to" part in our final answer.

How do I check if x ≥ 3 is correct?

Pick any number ≥ 3 (like x = 5) and substitute: $-2(5) + 9 = -1 \leq 3$ ✓. Also try the boundary: $-2(3) + 9 = 3 \leq 3$ ✓. Both work!

Why did the explanation say the answer is x ≤ 3 at the end?

That's actually an error in the explanation! The correct answer is $x \geq 3$ , as shown in all the work above. Always double-check the final statement matches your calculations.

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