Introduction to inequalities

Correct answer: A. (a\le c)

This is the transitive property of inequalities, so (a\le c) follows. Read the order carefully in chain inequalities.

Correct answer: D. (x\ge 9)

At least means (9) or more, so the inequality is (x\ge 9). In word problems connect at least with (\ge).

Correct answer: A. (x\le 3)

Subtracting (4) gives (-5x\ge -15), and dividing by (-5) gives (x\le 3). Reverse the sign when dividing by a negative number.

Correct answer: A. ((-4,1])

The value (-4) is not included so use an open bracket, and (1) is included so use a closed bracket. Read endpoint symbols carefully while writing intervals.

Correct answer: A. (t-6\le 14)

The statement says subtract (6) from (t) and use (\le), so it becomes (t-6\le 14). Do not change the order while translating words into symbols.

Correct answer: A. (-2\le x<4)

Dividing all parts by positive (3) gives (-2\le x<4). Inequality signs do not change when dividing by a positive number.

Correct answer: B. (7-2r>7-2s)

Multiplying (r<s) by (-2) reverses the sign to (-2r>-2s), and adding (7) keeps the direction. Check the sign at every step in mixed transformations.

Correct answer: A. (x\ge 3)

The direct answer is option A, \(x\ge 3\). Start with \(\frac{x+5}{4}\ge 2\). Since 4 is positive, multiply both sides by 4 without reversing the inequality: \(x+5\ge 8\). Subtract 5 from both sides: \(x\ge 3\). Thus every real number 3 or greater is a solution, including 3 itself. Option A is correct. Option B, \(x\le 3\), reverses the direction without justification. Option C, \(x\ge 13\), results from adding 5 instead of subtracting 5 after obtaining \(x+5\ge8\). Option D, \(x\le13\), contains both the wrong direction and the wrong number. The inequality sign would reverse only if we multiplied or divided by a negative number; the denominator 4 is positive. Memory cue: multiply by a positive number normally; multiply or divide by a negative number and reverse the inequality sign.

Correct answer: B. (x=3)

The solution is (3-4x<-9\Rightarrow -4x<-12\Rightarrow x>3), so (x=3) is not included. None of the listed values satisfies the inequality.

Correct answer: A. (2<x\leq 7)

Interval notation uses different brackets to show whether an endpoint is included. A round bracket means the endpoint is excluded, while a square bracket means the endpoint is included. Therefore the notation \((2,7]\) describes all real numbers greater than 2 and less than or equal to 7.

The left round bracket at 2 means \(x=2\) is not allowed, so the relation must be \(2<x\). The right square bracket at 7 means \(x=7\) is allowed, so the second relation is \(x\le7\). Combining them gives \(2<x\le7\), which is option A. Including 2 or excluding 7 would change the given interval.

Correct answer: A. (x\leq 2)

(-2x\geq -4), and dividing by a negative gives (x\leq 2). Reversing the sign is the key step here.

Correct answer: A. (x=4)

The direct answer is option A, x = 4. First solve the inequality: 2x + 1 < 9. Subtract 1 from both sides: 2x < 8. Divide by positive 2: x < 4. Thus every number smaller than 4 is a solution, but 4 itself is not included because the sign is strictly less than, not less than or equal to. Option A is therefore the value that is not a solution. Option B, x = 0, works because 2(0)+1 = 1 < 9. Option C, x = −2, works because −4+1 = −3 < 9. Option D, x = 3, works because 6+1 = 7 < 9. The boundary value is excluded in a strict inequality. Memory cue: a < sign leaves the endpoint out.

Correct answer: A. ({1,2,3,4})

Usually, (\mathbb{N}) means (1,2,3,\ldots). Thus the natural numbers less than (5) are ({1,2,3,4}).

Correct answer: C. \(x<6\)

The interval \(( -\infty,6)\) includes all real numbers less than \(6\), but does not include \(6\). That condition is exactly \(x<6\).

Correct answer: A. (-1<x\leq 4)

Subtracting (3) from all parts gives (-2<2x\leq 8), so (-1<x\leq 4). Handle both bounds together in compound inequalities.

Correct answer: A. Always true

The square of every real number is zero or positive. Therefore, x² ≥ 0 is true for every real x: x² = 0 when x = 0 and x² > 0 when x ≠ 0. Options C and D are incorrect because the statement is also true for negative and positive values, as well as for zero. Exam tip: When evaluating an inequality involving a square, always check the case x = 0.

Correct answer: A. (<)

(<) and (>) are strict inequalities because equality is not included. (\leq) and (\geq) include equality.

Correct answer: A. When both sides are multiplied by a negative number

Multiplying or dividing by a negative number reverses the inequality direction. This is the most common mistake in linear inequalities.

Correct answer: A. \(x = 3\)

In the inequality \(-2 < x \leq 3\), \(x\) must be greater than \(-2\) and less than or equal to 3. Thus, \(x=3\) is correct because 3 is included. \(x=-2\) is not allowed due to the strict inequality, while 4 and \(-3\) lie outside the interval. Exam tip: A strict sign \(<\) or \(>\) excludes the boundary, whereas \(\leq\) or \(\geq\) includes it.

Correct answer: A. ((-\infty,-1))

The inequality \(x<-1\) includes every real number smaller than -1, but it excludes -1 itself because the sign is strictly less than, not less than or equal to. Interval notation expresses this as \((-\infty,-1)\). The parenthesis at -1 shows that the endpoint is not included, and infinity always uses a parenthesis. Therefore option A is correct.

For example, -2 and -10 belong to this interval, while -1 does not. The interval \((-\infty,-1]\) would represent \(x\le-1\), which incorrectly includes -1. Intervals beginning at -1 represent values greater than or equal to -1 and therefore cannot describe the given inequality. Hence option A is the required representation.

Correct answer: A. (x\leq 4)

The symbol (\leq) is used for less than or equal to. Translating words into mathematical symbols is an important exam skill.

Correct answer: A. Open circle at (2) and shading to the right

The direct answer is A: an open circle at 2 with shading to the right. The sign “>” means values greater than 2. The number 2 itself is excluded, because the statement is not “greater than or equal to.” Therefore the circle at 2 must be open. Numbers greater than 2 lie to the right on a number line, so the ray or shading goes right. A shows both facts correctly. B has the correct direction but a closed circle, which would represent x≥2 and incorrectly include 2. C has an open circle, correctly excluding 2, but shades left; that represents x<2. D shades left and uses a closed circle, representing x≤2. A reliable method is to ask two questions: is the endpoint included, and in which direction are the allowed numbers? For “>”, use an open circle and right shading. Memory cue: greater goes right; strict signs use an open circle.

Correct answer: A. (x\geq 2)

To solve this linear inequality, first isolate the term containing x and then divide by its positive coefficient. Adding or subtracting the same number on both sides preserves the inequality direction. Dividing by a positive number also preserves it; only division by a negative number would reverse the sign.

From \(8\le2x+4\), subtract 4 from both sides to obtain \(4\le2x\). Divide both sides by 2, which is positive, and get \(2\le x\). Writing the variable first gives \(x\ge2\). Hence option A is correct. The equality is included because the original inequality uses \(\le\), not a strict inequality.

Correct answer: A. (x\leq 7)

Multiplying by positive (5) gives (2x+1\leq 15), so (x\leq 7). A positive denominator does not reverse the sign.

Correct answer: A. ([-1,4))

(-1) is included, so use a square bracket; (4) is not included, so use a round bracket. Check both endpoints separately while writing intervals.