Consider the compound disjunction: $3(x - 2) < 5x + 4$ or $4 - 2x \ge 10 - x$. Solve both inequalities and determine the complete solution set. Express the resulting set in interval notation.

Consider the compound disjunction: $3(x - 2) < 5x + 4$ or $4 - 2x \ge 10 - x$ . Solve both inequalities and determine the complete solution set. Express the resulting set in interval notation.

Final answer

$(-\infty, \infty)$

Steps

Solve the first inequality

To solve a compound disjunction joined by the word or, solve each linear inequality independently and then find the union of their solution sets.

$3(x - 2) < 5x + 4$

Distribute the $3$ across the parentheses: $3x - 6 < 5x + 4$
Subtract $3x$ from both sides to collect the variable terms on the right: $-6 < 2x + 4$
Subtract $4$ from both sides to isolate the variable term: $-10 < 2x$
Divide both sides by $2$ (since $2 > 0$ , the inequality direction remains unchanged): $-5 < x \quad \text{or equivalently,} \quad x > -5$

In interval notation, this solution is: $(-5, \infty)$

Solve the second inequality

$4 - 2x \ge 10 - x$

Add $2x$ to both sides to make the coefficient of $x$ positive: $4 \ge 10 + x$
Subtract $10$ from both sides: $-6 \ge x \quad \text{or equivalently,} \quad x \le -6$

(Alternative method: Subtracting $4$ gives $-2x \ge 6 - x$ , then adding $x$ gives $-x \ge 6$ . Dividing by $-1$ requires reversing the inequality sign, yielding $x \le -6$ .)

In interval notation, this solution is: $(-\infty, -6]$

Combine the solutions using the disjunction ("or")

A disjunction is satisfied by any value of $x$ that satisfies at least one of the two inequalities. Therefore, the complete solution set is the union of the two intervals: $(-\infty, -6] \cup (-5, \infty)$

Taking the union of the rays extending infinitely to the left and to the right across the real number line yields the complete unbounded solution set: $(-\infty, \infty)$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student treats the disjunction 'or' as a conjunction 'and' and looks for the intersection of the two solution sets, finding no overlap.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
3x - 6 < 5x + 4 $\implies$ -2x < 10 $\implies$ x > -5
Step 2
4 - 2x $\ge$ 10 - x $\implies$ -x $\ge$ 6 $\implies$ x $\le$ -6
Step 3
$\text{Intersection of }$ x > -5 $\text{ and }$ x $\le$ -6 $\text{ is empty}$ $\implies$ $\text{No solution}$
Correct
$(-\infty, -6] \cup (-5, \infty)$
A compound inequality joined by "or" represents a disjunction, requiring the union ( $\cup$ ) of the solution sets rather than their intersection ( $\cap$ ). Because any value of $x$ satisfying either inequality is a valid solution, the solution set is $(-\infty, -6] \cup (-5, \infty)$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded and solved the first inequality accurately to find $x > -5$ .

You correctly solved the second inequality, reversing the inequality symbol when dividing by a negative number to obtain $x \le -6$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Conjunction vs. Disjunction Confusion

You looked for the intersection of the two sets instead of taking their union.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You confused the logical connective 'or' (disjunction, corresponding to set union $\cup$ ) with 'and' (conjunction, corresponding to set intersection $\cap$ ).

REASONING PATH

Your ACEpath teacher’s suggested approach

Check the logical connective: An "or" compound inequality is satisfied by numbers in either set (union $\cup$ ), whereas an "and" compound inequality requires numbers to satisfy both sets simultaneously (intersection $\cap$ ).
Circle the word "or" in the prompt and write the $\cup$ symbol immediately to remind yourself to combine all valid intervals.

CORRECTED PATH

ACEpath Teacher's Demonstration

Express each inequality as an interval: $(-\infty, -6]$ and $(-5, \infty)$ .
Apply the "or" connective by taking the union of both intervals: $(-\infty, -6] \cup (-5, \infty)$ .

Final answer

$(-\infty, -6] \cup (-5, \infty)$

NEXT STEPS

Final Takeaways from Your Teacher

Practice distinguishing between 'and' (intersection) and 'or' (union) statements on a number line.
Review how to write disjoint intervals using the union symbol $\cup$ in interval notation.

OverviewACEpath Teacher's Diagnosis

02 The student forgets to reverse the inequality direction when dividing by a negative number in both inequalities.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
3x - 6 < 5x + 4 $\implies$ -2x < 10 $\implies$ x < -5
Correct
3x - 6 < 5x + 4 $\implies$ -2x < 10 $\implies$ x > -5
When you divided both sides of $-2x < 10$ by $-2$ , you kept the original $<$ symbol. Dividing by a negative value reverses the inequality sign, giving $x > -5$ .
Step 2
4 - 2x $\ge$ 10 - x $\implies$ -x $\ge$ 6 $\implies$ x $\ge$ -6
Correct
4 - 2x $\ge$ 10 - x $\implies$ -x $\ge$ 6 $\implies$ x $\le$ -6
When you divided both sides of $-x \ge 6$ by $-1$ , you maintained the $\ge$ symbol. Dividing by a negative value reverses the direction to $\le$ , yielding $x \le -6$ .
Step 3
x < -5 $\text{ or }$ x $\ge$ -6 $\text{ covers all real numbers}$ $\implies$ (- $\infty$ , $\infty$ )
Correct
x $\le$ -6 $\text{ or }$ x > -5 $\implies$ (- $\infty$ , -6] $\cup$ (-5, $\infty$ )
Because both component inequalities had reversed directions, your combined interval resulted in all real numbers instead of the disjoint union $(-\infty, -6] \cup (-5, \infty)$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly expanded the parentheses using the distributive property.

You correctly isolated the variable and constant terms on opposite sides.

You correctly evaluated the union of your own intermediate inequalities.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Inequality Direction Reversal Error

You performed the division by negative coefficients correctly in terms of magnitude, but you kept the original inequality directions instead of reversing them.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You missed the core rule of inequality arithmetic: multiplying or dividing both sides by a negative number inverts the order relation on the number line.

REASONING PATH

Your ACEpath teacher’s suggested approach

Division by a Negative Number: If $a < b$ and $c < 0$ , then $\frac{a}{c} > \frac{b}{c}$ .
Whenever you divide or multiply an inequality by a negative number, immediately flip the inequality sign.
Disjunction Union Verification: A disjunction ('or') takes the union of the individual solution sets.
Graph each individual ray on a single number line to clearly see if they overlap, leave a gap, or cover the entire line.

CORRECTED PATH

ACEpath Teacher's Demonstration

Divide $-2x < 10$ by $-2$ and flip the sign: $x > -5$ , which is $(-5, \infty)$ .
Divide $-x \ge 6$ by $-1$ and flip the sign: $x \le -6$ , which is $(-\infty, -6]$ .
Combine the two intervals using union: $(-\infty, -6] \cup (-5, \infty)$ .

Final answer

$(-\infty, -6] \cup (-5, \infty)$

NEXT STEPS

Final Takeaways from Your Teacher

Circle the negative coefficient in pencil whenever dividing to cue yourself to flip the inequality sign.
Test a test value in your solution set back into the original inequality to confirm that the direction is correct.

OverviewACEpath Teacher's Diagnosis

03 The student misinterprets the non-overlapping rays as a single bounded interval between -6 and -5.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
3x - 6 < 5x + 4 $\implies$ -2x < 10 $\implies$ x > -5
Step 2
4 - 2x $\ge$ 10 - x $\implies$ -x $\ge$ 6 $\implies$ x $\le$ -6
Step 3
-6, -5)
Correct
$(-\infty, -6] \cup (-5, \infty)$
The condition is a disjunction ("or"), meaning any number satisfying $x \le -6$ or $x > -5$ belongs to the solution. The interval $[-6, -5)$ represents numbers between $-6$ and $-5$ , which actually satisfy neither inequality. The correct representation is the union of the two rays: $(-\infty, -6] \cup (-5, \infty)$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded and simplified both linear inequalities with complete algebraic accuracy.

You remembered to reverse the inequality direction in both inequalities when dividing by negative numbers.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Misinterpreting disjunction of rays as an interior bounded interval

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated the boundary endpoints $-6$ and $-5$ as the lower and upper bounds of a single bounded interval, rather than sketching the two separate rays pointing outward toward negative and positive infinity.

REASONING PATH

Your ACEpath teacher’s suggested approach

Graph the rays before writing interval notation: For a compound disjunction ( $A \text{ or } B$ ), plot each ray on a number line and shade all points that lie in either ray.
Plot $x \le -6$ pointing left toward $-\infty$ and $x > -5$ pointing right toward $\infty$ . Since they point in opposite outward directions, write them as two disjoint intervals joined by the union symbol $\cup$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Identify the individual solution intervals: $x \le -6$ corresponds to $(-\infty, -6]$ , and $x > -5$ corresponds to $(-5, \infty)$ .
Combine the sets with the union operator $\cup$ for the compound 'or' statement: $(-\infty, -6] \cup (-5, \infty)$ .

Final answer

$(-\infty, -6] \cup (-5, \infty)$

NEXT STEPS

Final Takeaways from Your Teacher

Sketch both inequality solutions on a single number line to visually confirm which regions are included.
Verify interval notation by testing a sample value: for example, $x = 0$ satisfies $x > -5$ , so $0$ must be in the solution set, whereas $[-6, -5)$ excludes $0$ .](streamdown:incomplete-link)

Lock it in

Question

What set operation corresponds to a compound inequality joined by the word "or"?

Answer

Union ( $\cup$ ). Any number that makes at least one of the inequalities true is included in the solution set.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student treats the disjunction 'or' as a conjunction 'and' and looks for the intersection of the two solution sets, finding no overlap.

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 forgets to reverse the inequality direction when dividing by a negative number in both inequalities.

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 misinterprets the non-overlapping rays as a single bounded interval between -6 and -5.

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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