Solve the compound inequality $-13 \le 3 - 4x < 11$ for real numbers $x$. State the solution set in interval notation, and identify the smallest integer $x$ that satisfies the inequality.

Solve the compound inequality $-13 \le 3 - 4x < 11$ for real numbers $x$ . State the solution set in interval notation, and identify the smallest integer $x$ that satisfies the inequality.

Final answer

Interval: $(-2, 4]$ ; Smallest integer: $-1$

Steps

To solve the compound inequality $-13 \le 3 - 4x < 11$ , isolate the variable $x$ in the middle by applying standard inequality properties to all three parts simultaneously.

Step 1: Subtract $3$ from all three parts

-13 - 3 \le -4x < 11 - 3

-16 \le -4x < 8

Step 2: Divide all parts by $-4$

Key Rule and Common Pitfall: When dividing or multiplying an inequality by a negative number, you must reverse the inequality signs ( $\le$ becomes $\ge$ , and $<$ becomes $>$ ).

\frac{-16}{-4} \ge x > \frac{8}{-4}

4 \ge x > -2

Rewriting this in standard order from least to greatest:

-2 < x \le 4

Step 3: Express the solution in interval notation

Since $x$ is strictly greater than $-2$ (not including $-2$ ) and less than or equal to $4$ (including $4$ ), the solution set is:

(-2, 4]

Step 4: Identify the smallest integer satisfying the inequality

The integers belonging to the interval $(-2, 4]$ are:

\{-1, 0, 1, 2, 3, 4\}

Because $-2$ is excluded by the open boundary, the smallest integer value for $x$ is:

-1

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to reverse the inequality signs when dividing by a negative number, resulting in an reversed or inconsistent interval and selecting the wrong minimum integer.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
-13 $\le$ 3 - 4x < 11 $\implies$ -16 $\le$ -4x < 8
Step 2
4 $\le$ x < -2
Correct
4 $\ge$ x > -2 ( $\text{or }$ -2 < x $\le$ 4)
When dividing or multiplying an inequality by a negative number (here, $-4$ ), both inequality signs must flip directions: $\le$ becomes $\ge$ , and $<$ becomes $>$ .
Step 3
$\text{Solution set: } [4, -2)$ ](streamdown:incomplete-link)
Correct
(-2, 4]
This interval notation directly followed your unflipped inequality. Intervals must always be written with the smaller number on the left and the larger number on the right.
Step 4
$\text{Smallest integer is } 4$
Correct
-1
Because the interval was reversed, you selected 4 instead of looking for the smallest integer strictly greater than $-2$ , which is $-1$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly isolated the variable term $-4x$ in the middle by subtracting 3 from all three parts.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign reversal omission upon negative division

none

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You forgot the fundamental rule of inequalities: dividing or multiplying by a negative number reverses the order of the inequality signs.

REASONING PATH

Your ACEpath teacher’s suggested approach

Negative Multiplication / Division Rule: If $a < b$ and $c < 0$ , then $\frac{a}{c} > \frac{b}{c}$ .
Whenever you divide or multiply an inequality across by a negative coefficient, flip every inequality symbol immediately.
Sanity Check for Interval Order: Every real interval $[a, b]$ or $(a, b)$ must satisfy $a < b$ .
If your resulting boundaries appear as $[4, -2)$ , stop and check whether an inequality direction was missed, then rearrange the inequality from least to greatest.](streamdown:incomplete-link)

CORRECTED PATH

ACEpath Teacher's Demonstration

Divide each part of $-16 \le -4x < 8$ by $-4$ and flip both inequality symbols: $\frac{-16}{-4} \ge \frac{-4x}{-4} > \frac{8}{-4}$ , which simplifies to $4 \ge x > -2$ .
Rewrite the inequality from least to greatest: $-2 < x \le 4$ .
Write the solution set in interval notation: $(-2, 4]$ .
List the integers inside $(-2, 4]$ : $\{-1, 0, 1, 2, 3, 4\}$ . The smallest integer is $-1$ .

Final answer

Interval: $(-2, 4]$ ; Smallest integer: $-1$

NEXT STEPS

Final Takeaways from Your Teacher

Practice single-step and compound inequalities with negative coefficients to make flipping the sign an automatic habit.
Always write interval notation strictly in increasing order $(a, b]$ where $a < b$ , which instantly alerts you if an inequality was solved backwards.

OverviewACEpath Teacher's Diagnosis

02 The student confuses strict and non-strict inequality endpoints, treating the open boundary at -2 as inclusive and wrongly concluding that -2 is the smallest integer solution.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
-13 $\le$ 3 - 4x < 11 $\implies$ -16 $\le$ -4x < 8 $\implies$ 4 $\ge$ x > -2 $\implies$ -2 < x $\le$ 4
Step 2
$\text{Solution set: } [-2, 4]$
Correct
(-2, 4]
The inequality at the lower bound is strict ( $-2 < x$ ), which requires an open parenthesis $($ rather than a square bracket $[$ .](streamdown:incomplete-link)
Step 3
$\text{Integer solutions: }$ -2, -1, 0, 1, 2, 3, 4 $\implies$ $\text{Smallest integer is }$ -2
Correct
$\text{Integer solutions: }$ -1, 0, 1, 2, 3, 4 $\implies$ $\text{Smallest integer is }$ -1
Because of the bracket error in the previous step, you included $-2$ in the set of solutions. Since $x > -2$ , the smallest valid integer is $-1$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You isolated the variable term correctly by subtracting $3$ from each section.

You correctly flipped the inequality signs when dividing across by $-4$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Strict vs. Non-Strict Endpoint Confusion

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated the strict boundary $x > -2$ as inclusive, confusing the round parenthesis required for strict inequalities with the square bracket used for non-strict inequalities.

REASONING PATH

Your ACEpath teacher’s suggested approach

Strict vs. Non-Strict Intervals: Strict inequalities ( $<$ and $>$ ) exclude the boundary point and use parentheses $( )$ , whereas non-strict inequalities ( $\le$ and $\ge$ ) include the boundary point and use square brackets $[ ]$ .
Always inspect each endpoint symbol individually before choosing between a parenthesis and a bracket.
Integer Testing at Open Boundaries: An endpoint bounded by a strict inequality ( $x > a$ ) cannot be equal to $a$ .
Check whether the boundary value satisfies the original inequality: $-2 > -2$ is false, so $-2$ cannot be a solution.

CORRECTED PATH

ACEpath Teacher's Demonstration

Observe that $-2 < x$ means $-2$ is excluded, so the left endpoint must use a round parenthesis: $(-2$ .
Observe that $x \le 4$ means $4$ is included, so the right endpoint must use a square bracket: $4]$ .
Combine them into the interval $(-2, 4]$ .
List the integers strictly greater than $-2$ and up to $4$ : $\{-1, 0, 1, 2, 3, 4\}$ .
Select the smallest integer from the set: $-1$ .

Final answer

Interval: $(-2, 4]$ ; Smallest integer: $-1$

NEXT STEPS

Final Takeaways from Your Teacher

Review interval notation conventions: use parentheses $( )$ for $<$ and $>$ , and square brackets $[ ]$ for $\le$ and $\ge$ .
When finding the smallest or largest integer in an open interval, test the boundary value directly to confirm whether it satisfies the strict inequality.
Complete the three targeted flashcards and practice problems below to master interval endpoint boundaries.

OverviewACEpath Teacher's Diagnosis

03 The student adds 3 instead of subtracting 3 when isolating the x-term across the compound inequality.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
-13 $\le$ 3 - 4x < 11 $\implies$ -10 $\le$ -4x < 14
Correct
-13 - 3 $\le$ -4x < 11 - 3 $\implies$ -16 $\le$ -4x < 8
To eliminate the constant $+3$ in the middle expression $3 - 4x$ , you must subtract $3$ from each part of the compound inequality, rather than adding $3$ .
Step 2
2.5 $\ge$ x > -3.5 $\implies$ -3.5 < x $\le$ 2.5
Correct
$\frac{-16}{-4}$ $\ge$ x > $\frac{8}{-4}$ $\implies$ -2 < x $\le$ 4
Your division by $-4$ and sign reversal were executed correctly, but the values were shifted due to the arithmetic error in the previous step.
Step 3
$\text{Solution set: } (-3.5, 2.5]$
Correct
(-2, 4]
You correctly converted your compound inequality into interval notation with an open left boundary and closed right boundary, but the endpoints reflect the earlier error.
Step 4
$\text{Integer solutions: }$ -3, -2, -1, 0, 1, 2 $\implies$ $\text{Smallest integer is }$ -3
Correct
$\text{Integer solutions: }$ $\{$ -1, 0, 1, 2, 3, 4 $\}$ $\implies$ $\text{Smallest integer is }$ -1
You correctly identified the integers within your interval $(-3.5, 2.5]$ , but the true interval $(-2, 4]$ excludes $-2$ and has $-1$ as its smallest integer solution.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly recognized that dividing by a negative coefficient reverses both inequality symbols.

You properly converted strict and non-strict inequalities into interval notation using parentheses and square brackets.

You accurately listed the integers belonging to your computed interval and picked the minimum value.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Inverse Operation Error

You added 3 to each part instead of subtracting 3 to isolate the $-4x$ term.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the sign of the constant term $+3$ with the negative sign on $-4x$ , leading to applying addition instead of the required subtraction to undo $+3$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Identify the operation needed to cancel the constant: In an expression of the form $c - ax$ , the constant term $c$ is positive; undo it by subtracting $c$ , not adding $c$ .
Look at the sign in front of the constant term before choosing the operation to apply across all parts.

CORRECTED PATH

ACEpath Teacher's Demonstration

Subtract $3$ from all three parts: $-13 - 3 \le -4x < 11 - 3$ , giving $-16 \le -4x < 8$ .
Divide all parts by $-4$ and reverse the inequality signs: $4 \ge x > -2$ , which rewrites as $-2 < x \le 4$ .
Write the solution in interval notation: $(-2, 4]$ .
List the integer solutions in $(-2, 4]$ : $\{-1, 0, 1, 2, 3, 4\}$ .

Final answer

Interval: $(-2, 4]$ ; Smallest integer: $-1$

NEXT STEPS

Final Takeaways from Your Teacher

Check the sign of each isolated constant term carefully before deciding whether to add or subtract.
Always substitute a quick test value from your final interval back into the original inequality to catch arithmetic shifts early.

Lock it in

Question

What must you do to the inequality symbols whenever you multiply or divide all parts by a negative number?

Answer

Reverse every inequality symbol (e.g., $<$ becomes $>$ , and $\le$ becomes $\ge$ ).

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to reverse the inequality signs when dividing by a negative number, resulting in an reversed or inconsistent interval and selecting the wrong minimum integer.

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 confuses strict and non-strict inequality endpoints, treating the open boundary at -2 as inclusive and wrongly concluding that -2 is the smallest integer solution.

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 adds 3 instead of subtracting 3 when isolating the x-term across the compound inequality.

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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