Solve the Linear Inequality: 7x+c ≤ 4x-2

Q: What if I got x ≥ instead of x ≤?

Check your algebra steps! Most likely you either added instead of subtracted or flipped the sign incorrectly. Remember: we're moving terms to isolate x, and dividing by positive 3 keeps the ≤ direction.

Q: Does the answer depend on what c equals?

The boundary value depends on c, but the solution method stays the same! Whether c = 5 or c = -10, you'll always get the form $ x \leq -\frac{c+2}{3} $.

Linear Inequalities with Parameter Variables

Solve the inequality:

$7x+c \leq 4x-2$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the inequality:

$7x+c \leq 4x-2$

Step-by-step solution

To solve the inequality $7x+c \leq 4x-2$ , we follow these steps:

Subtract $4x$ from both sides: $3x + c \leq -2$
Subtract $c$ from both sides: $3x \leq -2 - c$
Divide both sides by $3$ : $x \leq -\frac{c+2}{3}$

Final Answer

$x \leq -\frac{c+2}{3}$

Key Points to Remember

Essential concepts to master this topic

Rule: Isolate variable terms on one side, constants on other
Technique: Subtract 4x from both sides: 7x - 4x = 3x
Check: Substitute boundary value into original inequality to verify direction ✓

Common Mistakes

Avoid these frequent errors

Flipping inequality sign when dividing by positive number
Don't flip the inequality when dividing 3x ≤ -2-c by positive 3 = wrong direction! The inequality sign only flips when multiplying or dividing by a negative number. Always keep the same direction when dividing by positive numbers.

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 doesn't the parameter c affect the inequality direction?

The parameter c only changes the boundary value, not the direction! Since we divide by positive 3, the ≤ sign stays the same. The inequality direction only flips when dividing by negative numbers.

How do I check my answer when there's a parameter?

Pick a specific value for c (like c = 1) and substitute both your solution and this c-value into the original inequality. Both sides should satisfy the inequality relationship.

What if I got x ≥ instead of x ≤?

Check your algebra steps! Most likely you either added instead of subtracted or flipped the sign incorrectly. Remember: we're moving terms to isolate x, and dividing by positive 3 keeps the ≤ direction.

Can I solve this without moving the parameter c?

No - you must treat c like any other number and move it to the constant side. The goal is always to get x by itself on one side of the inequality.

Does the answer depend on what c equals?

The boundary value depends on c, but the solution method stays the same! Whether c = 5 or c = -10, you'll always get the form $x \leq -\frac{c+2}{3}$ .

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