Introduction to inequalities

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

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

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

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.

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.

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

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.

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.

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.

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.

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

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.

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

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

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

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.

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

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

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.

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.

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

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.

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.

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

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