Inequalities & Absolute Value - ACT Math

Solve: $|x-6|>2$ (give your answer in interval notation).
Answer: $( -\infty,4)\cup(8,\infty)$ This absolute value inequality with > requires the exterior solution. |x - 6| > 2 means the distance from x to 6 is greater than 2. This gives x - 6 < -2 or x - 6 > 2. Solving: x < 4 or x > 8. In interval notation: (-∞, 4) ∪ (8, ∞) using parentheses since equality is excluded. Choice A shows the interior solution (for <), while choice C incorrectly includes the boundary points. For |expression| > constant, the solution excludes the boundary values and uses parentheses.
What is the range of $x$ if $|x - 5| < 3$ ?
Answer: $2 < x < 8$ This is an absolute value inequality of the form |expression| < constant, which represents distance. The inequality |x - 5| < 3 means the distance from x to 5 is less than 3. Split into the compound inequality: -3 < x - 5 < 3. Add 5 to all parts: 2 < x < 8. The solution is 2 < x < 8, which matches choice A. Choice B would be the solution if we had |x - 5| > 3, representing values outside the interval.
What is the range of $x$ if $|2x + 1| < 7$ ?
Answer: $-4 < x < 3$ This is an absolute value inequality involving a linear expression inside the absolute value. The inequality |2x + 1| < 7 means the distance from 2x + 1 to 0 is less than 7. Split into the compound inequality: -7 < 2x + 1 < 7. Subtract 1 from all parts: -8 < 2x < 6. Divide by 2: -4 < x < 3. The solution is -4 < x < 3, which matches choice A. Choice B would be the solution if we had |2x + 1| > 7.
Solve: $-5x + 3 < 8$
Answer: $x > -1$ This is a linear inequality with a negative coefficient requiring sign reversal. Starting with -5x + 3 < 8, subtract 3 from both sides to get -5x < 5. Divide by -5 (negative, so flip the inequality sign) to get x > -1. The solution is x > -1, which matches choice C. Choice B forgot to flip the inequality sign when dividing by a negative number, leading to the incorrect x < -1. Remember that dividing by negative values always requires flipping the inequality direction.
What is the solution set of the inequality $-3x + 2 > -7$ ?
Answer: $x < 3$ This linear inequality involves a negative coefficient, requiring attention to sign flips when dividing. Start with -3x + 2 > -7, then subtract 2 from both sides: -3x > -9. Divide both sides by -3 (negative number), so flip the inequality sign: x < 3. The solution is x < 3, meaning all values less than 3 satisfy the inequality. Choice B (x > 3) shows the error students make when they forget to flip the inequality sign after dividing by a negative coefficient. Always remember that dividing or multiplying by negative values reverses the inequality direction.
Solve: $4x + 5 < 13$
Answer: $x < 2$ This linear inequality involves isolating x through standard algebraic operations with positive coefficients. Start with 4x + 5 < 13, then subtract 5 from both sides: 4x < 8. Divide both sides by 4 (positive number, so no sign flip): x < 2. The solution is x < 2, meaning all values less than 2 satisfy the inequality. Choice C (x < 3) might result from arithmetic errors in the subtraction or division steps. Since we're working with positive coefficients throughout, the inequality direction remains unchanged at each step.
Which graph represents the solution to $-x + 5 > 3$ ?
Answer: Number line with shading from $2$ to the left, open circle at $2$ This is a linear inequality with a negative coefficient and graphical representation. Starting with -x + 5 > 3, subtract 5 from both sides to get -x > -2. Multiply by -1 (negative, so flip the inequality sign) to get x < 2. On a number line, this is represented by shading from 2 to the left with an open circle at 2. The solution matches choice C. Choice A would represent x > 2, which forgets the sign flip when dealing with the negative coefficient.
Which graph represents the solution to $2x + 1 \geq 5$ ?
Answer: Number line with shading from $2$ to the right, closed circle at $2$ This is a linear inequality requiring basic algebraic manipulation and graphical representation. Starting with 2x + 1 ≥ 5, subtract 1 from both sides to get 2x ≥ 4. Divide by 2 (positive, so no sign flip) to get x ≥ 2. On a number line, this is represented by shading from 2 to the right with a closed circle at 2 (since the inequality includes equality). The solution matches choice A. Choice C would represent x < 2, which is the opposite direction.
Which values of $x$ satisfy $2 < x - 1 < 5$ ?
Answer: $3 < x < 6$ This compound inequality requires adding a constant to all parts to isolate x. Start with 2 < x - 1 < 5, then add 1 to all three parts: 2 + 1 < x - 1 + 1 < 5 + 1. This simplifies to 3 < x < 6, meaning x is between 3 and 6 exclusive. The solution is 3 < x < 6, representing all values strictly between 3 and 6. Choice B (1 < x < 4) would result from subtracting 1 instead of adding 1 to isolate x. When solving compound inequalities, perform the same operation on all parts to maintain the relationships.
What is the solution set of the inequality $-2x+6>0$ ?
Answer: $x<3$ This is a linear inequality with a negative coefficient. Start with -2x + 6 > 0, then subtract 6 from both sides: -2x > -6. Divide both sides by -2 and flip the inequality sign (critical when dividing by negative): x < 3. The solution is all real numbers less than 3. Choice A incorrectly shows x > 3, likely from forgetting to flip the inequality sign when dividing by a negative number. Always remember: dividing or multiplying by a negative flips the inequality.
Which values of $x$ satisfy $5-x\ge 2$ ?
Answer: $x\le 3$ This linear inequality involves subtracting a variable term. Start with 5 - x ≥ 2, then subtract 5 from both sides: -x ≥ -3. Multiply both sides by -1 and flip the inequality sign: x ≤ 3. The solution includes all values less than or equal to 3. Choice A shows x ≥ 3, which would result from forgetting to flip the inequality when multiplying by -1. When isolating a variable with a negative coefficient, always remember to reverse the inequality sign.
What is the solution set of the inequality $3x - 5 > 4$ ?
Answer: $x > 3$ This is a linear inequality requiring basic algebraic manipulation. Starting with 3x - 5 > 4, add 5 to both sides to get 3x > 9. Divide both sides by 3 (positive, so no sign flip) to get x > 3. The solution is x > 3, which matches choice A. Choices C and D are mathematically equivalent to A and B respectively since 9/3 = 3, but they present the answer in an unnecessarily complex form. Always simplify fractions when possible for clarity.
Which graph represents the solution to $2x - 1 < 3$ ?
Answer: Number line with shading from $2$ to the left, open circle at $2$ This is a linear inequality requiring algebraic manipulation and graphical representation. Starting with 2x - 1 < 3, add 1 to both sides to get 2x < 4. Divide by 2 (positive, so no sign flip) to get x < 2. On a number line, this is represented by shading from 2 to the left with an open circle at 2 (since the inequality is strict). The solution matches choice A. Choice B would represent x ≥ -2, which is not our solution.
A quality check requires the error $x$ to be at least 3 units away from 2. Which values of $x$ satisfy $|x-2|\ge 3$ ? Give the solution set in interval notation.
Answer: $(-\infty,-1]\cup[5,\infty)$ The inequality |x-2|≥3 asks for values where the distance from 2 is at least 3. This splits into two cases: x-2≥3 or x-2≤-3. From the first case: x≥5. From the second case: x≤-1. The solution is x≤-1 or x≥5, which in interval notation is (-∞,-1]∪[5,∞). The brackets are closed because we have ≥ in the original inequality. Choice A would be the solution to |x-2|≤3, and choice B uses open parentheses which would mean > instead of ≥.
Which graph represents the solution to $x + 4 \leq 2$ ?
Answer: Number line with shading from $-2$ to the left, closed circle at $-2$ This is a linear inequality requiring algebraic manipulation and graphical representation. Starting with x + 4 ≤ 2, subtract 4 from both sides to get x ≤ -2. This means x can be -2 or any value less than -2. On a number line, this is represented by shading from -2 to the left with a closed circle at -2 (since the inequality includes equality). The solution matches choice A. Choice B shows shading from 6 to the left, which would represent x ≤ 6, not our solution.
What is the range of $x$ if $|x-3|<4$ ? (Write your answer as an inequality.)
Answer: $-1<x<7$ This absolute value inequality represents distance and splits into two cases. |x - 3| < 4 means the distance from x to 3 is less than 4. This gives us -4 < x - 3 < 4. Adding 3 to all parts: -1 < x < 7. The solution is all values strictly between -1 and 7. Choice A shows the exterior solution (for ≥), while choices C and D have incorrect boundaries. For |expression| < constant, always get the interior solution between two bounds.
Solve: $2(x-4)\ge 6$ .
Answer: $x\ge 7$ This inequality involves distribution and basic operations. Start with 2(x - 4) ≥ 6, then distribute: 2x - 8 ≥ 6. Add 8 to both sides: 2x ≥ 14. Divide by 2: x ≥ 7. The solution includes all values greater than or equal to 7. Choice A shows x ≥ 1, which would result from incorrect arithmetic (14 ÷ 2 = 7, not 1). Always double-check arithmetic when solving inequalities, as the algebraic rules remain the same as equations.
A student needs a score $x$ that is between 2 and 7, not including 2 but including 7. Which inequality is equivalent to the interval $ (2,7]$?
Answer: $2<x\le 7$ The interval notation (2,7] means all values greater than 2 but not including 2, up to and including 7. The parenthesis at 2 means x>2 (not x≥2), and the bracket at 7 means x≤7. Combined, this gives 2<x≤7. Choice A incorrectly includes 2 with ≥, choice C incorrectly excludes 7 with <, and choice D excludes both endpoints. When converting between interval and inequality notation, parentheses mean strict inequalities while brackets mean inclusive inequalities.
Solve: $5 - 2x \leq 1$
Answer: $x \geq 2$ This inequality requires isolating x while carefully handling the negative coefficient and inequality direction. Start with $5 - 2x \leq 1$ , then subtract 5 from both sides: $-2x \leq -4$ . Divide both sides by -2 (negative number), so flip the inequality sign: $x \geq 2$ . The solution is $x \geq 2$ , meaning all values greater than or equal to 2 satisfy the inequality. Students often forget to flip the inequality sign when dividing by negative numbers, leading to incorrect answers like choice B ( $x \leq 2$ ). Always remember that multiplying or dividing by negative values reverses the inequality direction.
Solve: $4 - 2x \geq 6$
Answer: $x \leq -1$ This is a linear inequality with a negative coefficient requiring sign reversal. Starting with $4 - 2x \geq 6$ , subtract 4 from both sides to get $-2x \geq 2$ . Divide by -2 (negative, so flip the inequality sign) to get $x \leq -1$ . The solution is $x \leq -1$ , which matches choice D. Choice B forgot to flip the inequality sign when dividing by a negative number, a common error. Always remember that dividing or multiplying by a negative flips the inequality direction.

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