representation on number line

Correct answer: A. ( -2,-1,0,1,2,3,4 )

( |x-1|\le 3 ) gives ( -2\le x\le 4 ). For integer solutions, take all integer points in this interval.

Correct answer: C. A closed dot at 2

The closed interval [2,2] contains only the number 2 because both endpoints are 2. An included endpoint is represented by a closed dot, so the correct representation is a closed dot at 2. An open dot would indicate that 2 is excluded. Exam tip: [a,a] always represents the singleton set containing only a.

Correct answer: A. ( x>5 ), open at ( 5 ), shaded right

Simplifying gives ( 4x+4>2x+14 ), so ( 2x>10 ) and ( x>5 ). With a strict sign, the dot remains open.

Correct answer: A. Closed circle at \(-1\), open circle at 1, and shading between them

Solve the double inequality in two parts. From \(-2<3-5x\), we get \(-5<-5x\); dividing by the negative number \(-5\) reverses the sign, giving \(x<1\). From \(3-5x\le8\), we get \(-5x\le5\), so \(x\ge-1\). Combining both conditions gives \(-1\le x<1\). Hence, \(-1\) is included but 1 is excluded. Option B reverses the inclusion of the two endpoints. Exam tip: always reverse an inequality sign when multiplying or dividing by a negative number.

Correct answer: C. Both open dots

Both signs are strict ( < ), so both boundary values are excluded. Use open dots for strict inequalities.

Correct answer: C. ( x\ge \frac{19}{2} ), closed at ( \frac{19}{2} ), shaded right

Multiplying by ( 3 ) gives ( 18\le 2x-1 ), so ( x\ge \frac{19}{2} ). The ( \le ) sign includes the boundary point.

Correct answer: C. ( \emptyset )

No real number can be less than ( 2 ) and greater than ( 8 ) at the same time. For AND, take the common part.

Correct answer: C. \(x\ge 14\), closed at \(14\), shaded right

In \(\frac{x-2}{-3}\le -4\), multiplying both sides by \(-3\) reverses the inequality sign: \(x-2\ge 12\). Hence, \(x\ge 14\). Therefore, the number line has a closed point at \(14\) and shading to the right. Option A incorrectly fails to reverse the inequality sign. Exam tip: always reverse an inequality when multiplying or dividing by a negative number.

Correct answer: C. Infinity is never a member of a solution set; therefore, an interval extending to infinity always uses a parenthesis at infinity

\(\infty\) and \(-\infty\) are not real numbers, so they cannot be included as members of an interval. Hence, intervals such as \((3,\infty)\) and \(( -\infty, 3]\) always use a parenthesis at infinity. A square bracket is used only when a real endpoint is included, so options A and B are incorrect. Exam tip: whenever infinity appears in interval notation, use a parenthesis next to it.

Correct answer: A. ( x<-8 ), open at ( -8 ), shaded left

Simplifying gives ( -10+2x>4x+6 ), so ( -16>2x ) and ( x<-8 ). In a strict inequality, the boundary point remains open.

Correct answer: B. ( x\le 2 ), closed dot at ( 2 ), shaded left

Multiplying by negative ( -5 ) reverses the sign and gives ( 3x+4\le 10 ). Therefore show ( x\le 2 ) with a closed dot.

Correct answer: D. ( (-\infty,6) )

The first part goes before ( -1 ), and the second includes ( -1 ) and goes before ( 6 ). So together they give all numbers less than ( 6 ).

Correct answer: C. ( [-4,4) )

The first inequality gives ( x\ge -4 ), and the second gives ( x<4 ). The common part of both conditions is ( [-4,4) ).