Solve the following multi-step linear inequality for $x$: $4(2x - 3) - 5x \ge 3(x + 4) - 8$. Express the solution in interval notation.

Solve the following multi-step linear inequality for $x$ : $4(2x - 3) - 5x \ge 3(x + 4) - 8$ . Express the solution in interval notation.

Final answer

No solution (or $\emptyset$ )

Steps

Expand both sides using the distributive property

To solve the inequality for $x$ , follow these algebraic steps:

Distribute the factors outside the parentheses across the terms inside:

4(2x - 3) - 5x &\ge 3(x + 4) - 8 \\ 8x - 12 - 5x &\ge 3x + 12 - 8 \end{aligned}$$

Combine like terms on each side

On the left-hand side, combine the variable terms $8x - 5x$ : $3x - 12$

On the right-hand side, combine the constant terms $12 - 8$ : $3x + 4$

Setting these back into the inequality gives: $3x - 12 \ge 3x + 4$

Eliminate the variable term from both sides

Subtract $3x$ from both sides to collect the variable terms:

3x - 12 - 3x &\ge 3x + 4 - 3x \\ -12 &\ge 4 \end{aligned}$$

Interpret the resulting statement

The variable $x$ has been eliminated, leaving the numerical comparison: $-12 \ge 4$

Since $-12$ is less than $4$ , this inequality is false for all real values of $x$ . A false statement independent of $x$ means there is no value of $x$ that can satisfy the original inequality.

Therefore, the inequality has no solution, which in set notation is represented as the empty set: $\emptyset$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student misidentifies a contradiction with eliminating variables as having all real numbers as solutions rather than no solution.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
8x - 12 - 5x $\ge$ 3x + 12 - 8 $\implies$ 3x - 12 $\ge$ 3x + 4
Step 2
$-12 \ge 4$
Step 3
$(-\infty, \infty)$
Correct
$\text{No solution (or } \emptyset \text{)}$
The statement $-12 \ge 4$ is mathematically false. When variable elimination leads to a false inequality (a contradiction), there is no value of $x$ that satisfies it, so the solution is no solution (or $\emptyset$ ), not $(-\infty, \infty)$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded the expressions using the distributive property without sign errors.

You combined variable and constant terms accurately on each side.

You subtracted $3x$ from both sides correctly to arrive at $-12 \ge 4$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Contradiction interpreted as identity

You solved all the algebra perfectly down to $-12 \ge 4$ , but you wrote $(-\infty, \infty)$ instead of recognizing that the false inequality means there are no solutions.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You confused a contradiction (an inequality that is always false, meaning no solution) with an identity (an inequality that is always true, meaning all real numbers).

REASONING PATH

Your ACEpath teacher’s suggested approach

Evaluate the constant inequality: After variables eliminate, ask: Is the remaining numerical statement true or false?
Check the numbers directly: $-12$ is less than $4$ , so $-12 \ge 4$ is false.
Match the truth value to the solution set: A false numerical statement means no real $x$ can make the inequality true.
If the statement is false, conclude 'no solution' or $\emptyset$ . If the statement is true (such as $-12 \le 4$ ), conclude $(-\infty, \infty)$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Compare the values on both sides: $-12$ is smaller than $4$ , so $-12 \ge 4$ is a false statement.
Since the statement is false for every value of $x$ , no real numbers satisfy the inequality.
Write the final solution set as no solution, or $\emptyset$ .

Final answer

No solution (or $\emptyset$ )

NEXT STEPS

Final Takeaways from Your Teacher

Practice checking whether variable-free inequalities are true or false on a number line.
Memorize the pair: True statement $\implies$ all real numbers $(-\infty, \infty)$ ; False statement $\implies$ no solution $\emptyset$ .

OverviewACEpath Teacher's Diagnosis

02 The student fails to distribute the multiplication to the constant term inside the parentheses, altering the constant comparison to a true inequality.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
8x - 3 - 5x $\ge$ 3x + 4 - 8
Correct
8x - 12 - 5x $\ge$ 3x + 12 - 8
When applying the distributive property $a(b + c) = ab + ac$ , you must multiply the outside factor by every term inside the parentheses. On the left side, $4(2x - 3) = 8x - 12$ (not $8x - 3$ ). On the right side, $3(x + 4) = 3x + 12$ (not $3x + 4$ ).
Step 2
3x - 3 $\ge$ 3x - 4
Correct
3x - 12 $\ge$ 3x + 4
You combined like terms correctly on both sides ( $8x - 5x = 3x$ and $4 - 8 = -4$ ), but the terms carried over the earlier distribution error.
Step 3
$-3 \ge -4$
Correct
$-12 \ge 4$
You correctly subtracted $3x$ from both sides, but the resulting numbers were affected by the earlier distribution mistake.
Step 4
$(-\infty, \infty)$
Correct
$\emptyset$
While $-3 \ge -4$ is indeed always true, the correct simplified inequality is $-12 \ge 4$ , which is a contradiction (false for all $x$ ). Therefore, the true solution set is the empty set, $\emptyset$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly combined the variable terms and constant terms on each side of the inequality.

You correctly eliminated the variable term from both sides by subtracting $3x$ .

You correctly recognized that a universally true statement like $-3 \ge -4$ corresponds to $(-\infty, \infty)$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributive Property Incomplete

You only multiplied the outside factor by the variable term and left the constant term unmultiplied on both sides.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Incomplete application of the distributive property across grouped binomials, leading to an incorrect constant comparison and reversing the truth value of the final numerical statement.

REASONING PATH

Your ACEpath teacher’s suggested approach

Full Distribution Check: Every term inside parentheses must be multiplied by the factor outside: $a(b + c) = ab + ac$ .
Draw distribution arrows or underline each internal term before removing parentheses to verify that the constants are also multiplied.
Truth Value Interpretation: When variables cancel out completely: a true numerical statement (e.g., $4 \ge 0$ ) means $(-\infty, \infty)$ , while a false statement (e.g., $-12 \ge 4$ ) means no solution ( $\emptyset$ ).
Always double-check the arithmetic of constant terms when variables cancel out entirely.

CORRECTED PATH

ACEpath Teacher's Demonstration

Combine like terms on each side: $3x - 12 \ge 3x + 4$ .
Subtract $3x$ from both sides: $-12 \ge 4$ .
Observe that $-12 \ge 4$ is a false statement for all real numbers.

Final answer

No solution (or $\emptyset$ )

NEXT STEPS

Final Takeaways from Your Teacher

Practice expanding binomials with explicit distribution arrows to avoid dropping the constant multiplication.
Solve linear inequalities where the variable terms cancel out, paying special attention to whether the final constant inequality is true or false.

OverviewACEpath Teacher's Diagnosis

03 The student forgets to apply the negative sign when distributing across the second term inside the parentheses, resulting in a false condition that keeps a variable term.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
8x + 12 - 5x $\ge$ 3x + 12 - 8
Correct
8x - 12 - 5x $\ge$ 3x + 12 - 8
When distributing $4$ across $(2x - 3)$ , multiplying $4$ by $-3$ yields $-12$ , not $+12$ .
Step 2
3x + 12 $\ge$ 3x + 4
Correct
3x - 12 $\ge$ 3x + 4
You combined like terms correctly on both sides ( $8x - 5x = 3x$ and $12 - 8 = 4$ ), but the $+12$ carried forward from the expansion error.
Step 3
$12 \ge 4$
Correct
$-12 \ge 4$
You correctly subtracted $3x$ from both sides, but the remaining constant on the left should be $-12$ , not $12$ .
Step 4
$(-\infty, \infty)$
Correct
$\text{No solution (or } \emptyset \text{)}$
Because the true inequality is $-12 \ge 4$ , which is false for all real numbers, there is no solution rather than all real numbers.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly applied the distributive property to the right-hand expression $3(x + 4)$ .

You combined variable and constant terms on both sides accurately.

You correctly eliminated the variable term $3x$ from both sides.

You correctly understood that a universally true numerical statement corresponds to $(-\infty, \infty)$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributive Property Sign Error

You wrote $+12$ instead of $-12$ when multiplying $4 \times (-3)$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking the negative sign inside $(2x - 3)$ during distribution reversed the sign of the constant term, converting a false statement ( $-12 \ge 4$ ) into a true statement ( $12 \ge 4$ ).

REASONING PATH

Your ACEpath teacher’s suggested approach

Sign Tracking in Distribution: $a(b - c) = ab - ac$
Circle the sign in front of each term inside parentheses before multiplying.
Special Case Interpretation: False statement without variables $\implies \emptyset$ ; True statement without variables $\implies (-\infty, \infty)$
Always double-check the constant terms whenever all variable terms cancel out.

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute $4$ correctly on the left: $4(2x - 3) - 5x = 8x - 12 - 5x$ .
Distribute $3$ on the right: $3(x + 4) - 8 = 3x + 12 - 8$ .
Combine like terms: $3x - 12 \ge 3x + 4$ .
Subtract $3x$ from both sides: $-12 \ge 4$ .
Notice that $-12 \ge 4$ is false for all $x$ .

Final answer

$\text{No solution (or } \emptyset\text{)}$

NEXT STEPS

Final Takeaways from Your Teacher

Practice distributing over subtraction by writing out the multiplication of each term explicitly, e.g., $4(2x) + 4(-3)$ .
Review special solutions of linear inequalities: recognize when variable elimination yields an identity versus a contradiction.

Lock it in

Question

When solving an inequality and all variables cancel out, how do you determine the solution?

Answer

Look at the remaining numerical statement: if it is true, the solution is all real numbers, $(-\infty, \infty)$ ; if it is false, there is no solution, $\emptyset$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student misidentifies a contradiction with eliminating variables as having all real numbers as solutions rather than no solution.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student fails to distribute the multiplication to the constant term inside the parentheses, altering the constant comparison to a true inequality.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student forgets to apply the negative sign when distributing across the second term inside the parentheses, resulting in a false condition that keeps a variable term.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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