representation on number line

Correct answer: B. Closed dot at \(5\), shaded right

Adding \(11\) to both sides of \(4x-11\ge 9\) gives \(4x\ge 20\). Dividing by \(4\) gives \(x\ge 5\). Thus, \(5\) is included in the solution, so the point at \(5\) is closed and the shading extends to the right for values greater than \(5\). Option A represents \(x>5\) because it has an open dot, so it is incorrect. Exam tip: use a closed dot for \(\ge\) or \(\le\), and an open dot for \(>\) or \(<\).

Correct answer: C. ( -3,-2,-1,0,1,2,3 )

( -3 ) is included and ( 4 ) is excluded. For integer solutions, mark separate points instead of a continuous shade.

Correct answer: B. ( (-\infty,-1]\cup(7,\infty) )

In the complement, ( -1 ) becomes included and ( 7 ) is excluded. Open and closed endpoints switch when taking complements.

Correct answer: C. ( x\le -4 ) or ( x>3 )

The first part is ( x\le -4 ), and the second is ( x>3 ). Separate shaded parts are usually joined by OR.

Correct answer: A. \([2,\infty)\)

In \([2,\infty)\), 2 is included, so it is the least element. The interval extends indefinitely to the right, so no greatest element exists. Exam tip: a square bracket means that endpoint is included.

Correct answer: C. ( [-2,10] )

( |x-4|\le 6 ) gives ( -6\le x-4\le 6 ). Hence both endpoints are included and the interval is ( [-2,10] ).

Correct answer: A. ( x<-4 ) or ( x>3 )

( |2x+1|>7 ) gives ( 2x+1<-7 ) or ( 2x+1>7 ). Because the sign is strict, ( -4 ) and ( 3 ) remain open.

Correct answer: B. ( [-2,6) )

The first inequality gives ( x<6 ), and the second gives ( x\ge -2 ). For both conditions, the common part is ( [-2,6) ).

Correct answer: A. \(-3 < x < 4\)

In \(-3 < x < 4\), x takes all values between \(-3\) and \(4\), but neither endpoint is included. Hence, the number line has open circles at \(-3\) and \(4\), with the segment between them shaded; this is a bounded open interval. Option B includes both endpoints, so it represents a closed interval. Exam tip: Use an open circle for ‘<’ or ‘>’ and a filled circle for ‘≤’ or ‘≥’.

Correct answer: C. No solution

No number can be both greater than ( 4 ) and less than or equal to ( 4 ). For AND, the intersection is empty.

Correct answer: B. Only a closed dot at \(-3\)

An intersection contains only values that satisfy both inequalities at the same time. Both \(x\le -3\) and \(x\ge -3\) are true only for \(x=-3\). Hence, the number line shows a closed dot at \(-3\). Option A includes values less than \(-3\), and option C includes values greater than \(-3\), so neither represents the intersection. Exam tip: When opposite inequalities have the same boundary value, the intersection is often that single boundary point.

Correct answer: C. Open dot at ( \frac{5}{3} ), shaded left

The round bracket excludes ( \frac{5}{3} ), and ( -\infty ) indicates the left direction. Infinity never gets a closed endpoint.

Correct answer: A. ( -\frac{7}{4} ) closed and ( \frac{1}{2} ) open

The endpoint with ( \le ) is closed and the endpoint with ( < ) is open. Place fractions in order on the number line.

Correct answer: A. ( x>2 ), open at ( 2 ), shaded right

Solving gives ( -3x<-6 ), so ( x>2 ). Reverse the inequality sign when dividing by a negative number.

Correct answer: D. ( (-1,6) )

Both endpoints are open, so round brackets are used on both sides. Open dots do not include boundary values.

Correct answer: A. \([-11,7)\)

Multiplying all three parts of the inequality by the positive number 3 gives \(-6\le x+5<12\). Subtracting 5 from all three parts gives \(-11\le x<7\). Therefore, \(-11\) is included, so it takes a square bracket, while \(7\) is not included, so it takes a parenthesis: \([-11,7)\). In option B, the inclusion status of both endpoints is reversed. Exam tip: In a compound inequality, track each inequality sign—\(\le\), \(<\), \(\ge\), or \(>\)—separately.

Correct answer: B. ( \emptyset )

No number can satisfy ( 2<x\le 2 ). With equal boundaries and conflicting endpoint conditions, the interval is empty.

Correct answer: A. ( \mathbb{R} )

The first part includes ( 0 ), and the second gives values greater than ( 0 ). Together they form the whole real line.

Correct answer: A. ( x\ge -1 )

Simplifying gives ( 6-3x\le x+10 ), so ( -4x\le 4 ) and ( x\ge -1 ). The sign reverses when dividing by a negative coefficient.

Correct answer: B. ( \left(-\frac{3}{2},\frac{9}{2}\right] )

The left side is strict, so use a round bracket, and the right side includes equality, so use a square bracket. Write the smaller value first in interval notation.

Correct answer: A. ( (-\infty,-6)\cup[-2,\infty) )

The given interval includes ( -6 ) and excludes ( -2 ). In the complement, ( -6 ) is removed and ( -2 ) is included.

Correct answer: A. ( [-5,-3) )

Solving gives ( 6<-2x\le 10 ), then signs reverse to ( -3>x\ge -5 ). In order, write ( -5\le x<-3 ).

Correct answer: C. Open at ( -\frac{2}{3} ), shaded right

The sign ( > ) excludes the boundary, and greater values lie to the right. Apply the same rule for fractional boundaries.

Correct answer: C. ( x\le -1 ), closed at ( -1 ), shaded left

Multiplying by ( -2 ) reverses the sign and gives ( 7-x\le -8 ). Then ( -x\le -15 ), so ( x\ge 15 ).

Correct answer: C. ( -1<x\le 9 )

The open dot excludes ( -1 ), and the closed dot includes ( 9 ). In exams, first identify the type of both endpoints.