Solve the linear inequality $-3(2x - 5) \ge 4x - 25$ for $x$, express the solution set in interval notation, and describe the geometric representation of the solution set on a real number line.

Solve the linear inequality $-3(2x - 5) \ge 4x - 25$ for $x$ , express the solution set in interval notation, and describe the geometric representation of the solution set on a real number line.

Final answer

Interval notation: $(-\infty, 4]$ ; on the number line, a solid closed circle at $x = 4$ with a shaded ray extending indefinitely to the left.

Steps

Expand and simplify both sides

To solve the inequality, write the solution in interval notation, and graph it on a number line, follow these steps:

Apply the distributive property to the left side of the inequality: $-3(2x - 5) \ge 4x - 25$ $-6x + 15 \ge 4x - 25$

Collect variable terms and constant terms

Subtract $4x$ from both sides to group the $x$ -terms on the left: $-6x - 4x + 15 \ge -25$ $-10x + 15 \ge -25$

Subtract $15$ from both sides to isolate the variable term: $-10x \ge -25 - 15$ $-10x \ge -40$

Divide by the coefficient of $x$

Divide both sides by $-10$ .

Important Rule: Whenever an inequality is multiplied or divided by a negative number, the direction of the inequality sign must be reversed.

$\frac{-10x}{-10} \le \frac{-40}{-10}$ $x \le 4$

Write in interval notation

The inequality $x \le 4$ includes all real numbers less than or equal to $4$ .

The lower bound extends indefinitely to the left: $-\infty$ (always written with an open parenthesis ().
The upper bound is $4$ , which is included in the solution set (written with a closed bracket ]).

Therefore, the interval notation is: $(-\infty, 4]$

Describe the graph on a number line

To represent $x \le 4$ geometrically on a horizontal number line:

Place a solid (closed) circle at the point $4$ to indicate that $4$ is included in the solution set.
Shade a continuous ray pointing to the left from $4$ , ending with an arrow pointing toward $-\infty$ , representing all values smaller than $4$ .

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to reverse the inequality direction when dividing by a negative number, resulting in x ≥ 4.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
-3(2x - 5) $\ge$ 4x - 25 $\implies$ -6x + 15 $\ge$ 4x - 25 $\implies$ -10x $\ge$ -40
Step 2
$x \ge 4$
Correct
$x \le 4$
When dividing or multiplying both sides of an inequality by a negative number (here, $-10$ ), the inequality sign must flip direction. Dividing by $-10$ changes $\ge$ to $\le$ , yielding $x \le 4$ .
Step 3
$[4, \infty)$ ](streamdown:incomplete-link)
Correct
$(-\infty, 4]$
Your interval notation $[4, \infty)$ correctly reflects your intermediate result $x \ge 4$ , but because the sign was not reversed in the previous step, the true solution set is $(-\infty, 4]$ .
Step 4
$\text{A solid circle at }$ 4 $\text{ with shading extending to the right}$
Correct
$\text{A solid circle at }$ 4 $\text{ with shading extending to the left toward }$ - $\infty$
Your graph accurately matches your written interval $[4, \infty)$ , but since the actual solution is $x \le 4$ , the ray should shade to the left toward $-\infty$ .](streamdown:incomplete-link)

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded $-3(2x - 5)$ accurately to get $-6x + 15$ .

You grouped variable terms and constant terms cleanly to isolate $-10x \ge -40$ .

You correctly used a solid closed circle at $4$ to represent an inclusive endpoint.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign direction reversal omission

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated division by a negative coefficient just like division in an equation, overlooking the fundamental inequality rule that dividing or multiplying by a negative number reflects values across zero and reverses their relative order.

REASONING PATH

Your ACEpath teacher’s suggested approach

Check the divisor's sign: Whenever you multiply or divide an inequality by a negative quantity, the inequality symbol must immediately flip (e.g., $\ge$ becomes $\le$ ).
Circle negative coefficients before dividing so you remember to flip the inequality sign in the very next line.
Test a test point: Any valid solution from your final set must satisfy the original inequality.
Pick an easy test point such as $x = 0$ . In the original inequality, $-3(0 - 5) = 15 \ge -25$ is true, so $0$ must be in the shaded region. Your graph shaded values greater than $4$ , leaving out $0$ , which signals an inverted sign.

CORRECTED PATH

ACEpath Teacher's Demonstration

Divide both sides of $-10x \ge -40$ by $-10$ and reverse the inequality symbol: $x \le 4$ .
Write $x \le 4$ in interval notation: $(-\infty, 4]$ .
Plot on the number line: place a solid closed circle at $4$ and shade the ray extending to the left toward $-\infty$ .

Final answer

Interval notation: $(-\infty, 4]$ ; Number line: a solid closed circle at $4$ with a shaded ray extending to the left.

NEXT STEPS

Final Takeaways from Your Teacher

Practice solving 1-step inequalities involving negative division to build automaticity with sign reversal.
Form the habit of testing $x = 0$ in the original problem to verify whether the shaded direction is correct.

OverviewACEpath Teacher's Diagnosis

02 The student incorrectly distributes the negative sign to the second term inside the parentheses as -15 instead of +15, leading to -10x ≥ -10 and x ≤ 1.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
-3(2x - 5) $\ge$ 4x - 25 $\implies$ -6x - 15 $\ge$ 4x - 25
Correct
-3(2x - 5) $\ge$ 4x - 25 $\implies$ -6x + 15 $\ge$ 4x - 25
When distributing $-3$ to both terms inside the parentheses, multiplying two negative numbers gives a positive: $(-3) \times (-5) = +15$ , not $-15$ .
Step 2
-10x $\ge$ -10 $\implies$ x $\le$ 1
Correct
-10x $\ge$ -40 $\implies$ x $\le$ 4
Your algebraic rearrangement and rule application were correct based on your previous step (including reversing the inequality symbol when dividing by $-10$ ), but the values were affected by the earlier sign mistake.
Step 3
(- $\infty$ , 1] $\text{ and a solid dot at }$ 1 $\text{ shaded to the left}$
Correct
(- $\infty$ , 4] $\text{ and a solid dot at }$ 4 $\text{ shaded to the left}$
Your interval notation and number line representation accurately matched your value of $x \le 1$ , but the correct endpoint is $4$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly remembered to reverse the direction of the inequality sign when dividing both sides by the negative coefficient $-10$ .

You correctly represented a non-strict inequality ( $\le$ ) using a closed bracket in interval notation and a solid circle on the number line.

You correctly shaded the number line toward negative infinity to represent values less than the endpoint.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributive Property Sign Error

Multiplying $-3$ by $-5$ yielded $-15$ instead of $+15$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking the interaction between the negative factor outside parentheses and the negative sign on the subtracted term inside parentheses.

REASONING PATH

Your ACEpath teacher’s suggested approach

Distribute Signs Carefully: For any real numbers $a$ , $b$ , and $c$ , $-a(b - c) = -ab + ac$ .
Always circle or highlight the sign attached to each term inside the parentheses before distributing a negative multiplier.

CORRECTED PATH

ACEpath Teacher's Demonstration

$-3(2x) - 3(-5) \ge 4x - 25 \implies -6x + 15 \ge 4x - 25$
Subtract $4x$ and $15$ from both sides: $-10x \ge -40$
Divide by $-10$ and flip the inequality sign: $x \le 4$

Final answer

Interval notation: $(-\infty, 4]$ ; on the number line, a solid closed circle at $x = 4$ with a shaded ray extending to the left.

NEXT STEPS

Final Takeaways from Your Teacher

Review multiplying signed numbers, paying special attention to $(-)(-) = (+)$ .
Practice expanding parentheses where a negative constant multiplies a difference, such as $-a(x - b)$ .
Substitute a test point (such as $x = 0$ ) into the original inequality to check whether your final solution set satisfies it.

OverviewACEpath Teacher's Diagnosis

03 The student confuses non-strict and strict inequalities at the boundary, treating ≥ as > and using an open interval and an open circle at 4.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
$-3(2x - 5) \ge 4x - 25 \implies -6x + 15 \ge 4x - 25 \implies -10x \ge -40$
Step 2
$x < 4$
Correct
$x \le 4$
When dividing both sides of $-10x \ge -40$ by $-10$ , the inequality symbol reverses from $\ge$ to $\le$ . You correctly flipped the direction, but omitted the 'or equal to' bar, treating a non-strict inequality as a strict one.
Step 3
$(-\infty, 4)$
Correct
$(-\infty, 4]$
Because $x \le 4$ includes the boundary point $4$ , the upper bound requires a square bracket ] rather than a parenthesis ).
Step 4
An open circle at $x = 4$ with a shaded ray extending to the left.
Correct
A solid (closed) circle at $x = 4$ with a shaded ray extending indefinitely to the left.
Because $4$ is included in the solution set ( $x \le 4$ ), the point at $4$ must be plotted as a solid closed circle.

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded $-3(2x - 5)$ accurately using the distributive property.

You collected like terms correctly to reach $-10x \ge -40$ .

You correctly remembered to reverse the direction of the inequality when dividing by the negative coefficient $-10$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Strict vs. Non-Strict Inequality Boundary Confusion

You wrote $<$ instead of $\le$ when dividing both sides by $-10$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You focused on reversing the inequality direction upon dividing by a negative number, but lost track of whether the original inequality was strict or non-strict, unintentionally dropping the equality bar.

REASONING PATH

Your ACEpath teacher’s suggested approach

Preserve Equality Status: Dividing or multiplying by a negative number reverses the direction of the inequality, but does not alter whether equality is included: $\ge$ becomes $\le$ , and $>$ becomes $<$ .
Always copy the horizontal equality bar along with the flipped inequality sign unless you are intentionally splitting cases.
Match Symbols to Notation: Non-strict inequalities ( $\le, \ge$ ) always pair with square brackets [ or ] and solid closed circles on the number line.
Before writing interval notation or drawing a number line, inspect the inequality sign: if it has an equality bar, use a bracket and fill in the circle.

CORRECTED PATH

ACEpath Teacher's Demonstration

Divide both sides by $-10$ and reverse $\ge$ to $\le$ : $x \le 4$ .
Express all values less than or equal to $4$ in interval notation: $(-\infty, 4]$ .
Graph the solution by plotting a solid closed circle at $4$ and drawing a ray pointing left toward $-\infty$ .

Final answer

Interval notation: $(-\infty, 4]$ ; on the number line, a solid closed circle at $x = 4$ with a shaded ray extending indefinitely to the left.

NEXT STEPS

Final Takeaways from Your Teacher

Review the correspondence table: $\le / \ge \longleftrightarrow [\ /\ ] \longleftrightarrow \text{solid dot}$ ; $< / > \longleftrightarrow (\ /\ ) \longleftrightarrow \text{open dot}$ .
Practice two-step inequalities where division by a negative coefficient occurs on a non-strict inequality to build the habit of keeping the equality bar intact.

Lock it in

Question

What essential rule must you apply whenever you multiply or divide both sides of an inequality by a negative number?

Answer

Reverse the direction of the inequality sign (e.g., $>$ becomes $<$ , and $\ge$ becomes $\le$ ).

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to reverse the inequality direction when dividing by a negative number, resulting in x ≥ 4.

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 incorrectly distributes the negative sign to the second term inside the parentheses as -15 instead of +15, leading to -10x ≥ -10 and x ≤ 1.

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 confuses non-strict and strict inequalities at the boundary, treating ≥ as > and using an open interval and an open circle at 4.

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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