Solve linear inequalities

1. The variable has 7 added to it, so do the opposite: subtract 7 from both sides. x + 7 - 7 > 12 - 7
2. Simplify each side: x > 5
3. Subtracting never flips the inequality sign, and the symbol stays strict (>, not \ge).

Answer: x > 5

Check: Try x = 6: 6 + 7 = 13 and 13 > 12 ✓. Try the boundary x = 5: 5 + 7 = 12 and 12 > 12 is false, so 5 is correctly not included.

Tip: Adding or subtracting the same number on both sides never changes the direction of the inequality — only multiplying or dividing by a negative does.

1. The coefficient of x is -4. Divide both sides by -4.
2. Because you are dividing by a negative number, reverse the sign: \le becomes \ge. \frac{-4x}{-4} \ge \frac{20}{-4}
3. Simplify: x \ge -5

Answer: x \ge -5

Check: Try x = 0 (in the solution set): -4(0) = 0 and 0 \le 20 ✓. Try x = -10 (not in the set): -4(-10) = 40 and 40 \le 20 is false ✓. The boundary x = -5 gives 20 \le 20, true, so the circle is closed.

Tip: Whenever the number in front of the variable is negative, write the flipped sign on your erasable note board before you do the division.

1. 9 is subtracted from x, so do the opposite: add 9 to both sides. x - 9 + 9 \ge -14 + 9
2. Simplify. Adding a positive number to -14 moves 9 units to the right on the number line: x \ge -5
3. You added a number, so the sign does not change. The negative constant on the right side does not matter: only multiplying or dividing by a negative number flips the sign.

Answer: x \ge -5

Check: Boundary x = -5: -5 - 9 = -14, and -14 \ge -14 is true, so -5 is included. Inside the set, x = 0: 0 - 9 = -9 \ge -14 ✓. Outside, x = -6: -15 \ge -14 is false ✓.

Tip: A negative number sitting on the right side is never a reason to flip. Flip only when you multiply or divide both sides by a negative.

1. x is divided by -5, so undo it by multiplying both sides by -5.
2. Multiplying by a negative number reverses the sign: < becomes >. \frac{x}{-5} \cdot (-5) > 3 \cdot (-5)
3. Simplify: x > -15

Answer: x > -15

Check: Try x = 0 (in the set): \frac{0}{-5} = 0 and 0 < 3 ✓. Try x = -20 (not in the set): \frac{-20}{-5} = 4 and 4 < 3 is false ✓. The boundary x = -15 gives 3 < 3, false, so the sign stays strict.

Tip: Undoing a negative denominator is a multiplication by a negative, so it flips the sign exactly like dividing by a negative does.

1. x is divided by 4, so multiply both sides by 4.
2. You are multiplying by positive 4, so the sign stays >. (The -2.5 is only a number on one side; it does not cause a flip.) \frac{x}{4} \cdot 4 > -2.5 \cdot 4
3. Simplify: x > -10

Answer: x > -10

Check: Try x = 0: \frac{0}{4} = 0 > -2.5 ✓. Try x = -12: \frac{-12}{4} = -3 and -3 > -2.5 is false ✓ (-3 is further left). The boundary x = -10 gives -2.5 > -2.5, false, so the sign stays strict.

Tip: Ask "what am I multiplying or dividing by?" — only a negative multiplier or divisor flips the sign. The signs of the other numbers do not matter.

1. Subtract 5 from both sides (the 5 is positive, so subtract it): 5 - 2x - 5 > 11 - 5 \quad\Rightarrow\quad -2x > 6
2. Divide both sides by -2. Dividing by a negative reverses the sign, so > becomes <: x < \frac{6}{-2}
3. Simplify: x < -3

Answer: x < -3

Check: Try x = -4: 5 - 2(-4) = 5 + 8 = 13 and 13 > 11 ✓. Try x = 0: 5 > 11 is false ✓, so values to the right of -3 really are excluded.

Tip: The term -2x carries its minus sign with it. Treat it as "-2 times x", which is why the last step is a division by -2.

1. Distribute the 2 to both terms in the parentheses: 2x + 8 \ge 3x - 1
2. Subtract 2x from both sides (this keeps the x-term positive): 8 \ge x - 1
3. Add 1 to both sides: 9 \ge x
4. Read it from right to left, turning the symbol with it: 9 \ge x means x \le 9.

Answer: x \le 9

Check: Boundary x = 9: 2(13) = 26 and 3(9) - 1 = 26, equal ✓. Inside, x = 0: 8 \ge -1 ✓. Outside, x = 10: 28 \ge 29 is false ✓.

Tip: Collect the variable on the side where its coefficient stays positive — then you never have to flip.

1. Add 1 to both sides: \frac{2}{3}x < 8
2. Multiply both sides by the reciprocal \dfrac{3}{2} (positive, no flip): x < 8 \cdot \frac{3}{2} = \frac{24}{2} = 12

Answer: x < 12

Check: Boundary x = 12: \tfrac{2}{3}(12) - 1 = 8 - 1 = 7, equal to the right side ✓. Inside, x = 6: 4 - 1 = 3 < 7 ✓.

Tip: To undo multiplying by a fraction, multiply by its reciprocal (flip the fraction upside down). This "flip" of the fraction is not the same as flipping the inequality sign — the sign only flips for negatives.

1. Subtract 9 from both sides: 9 - 4x - 9 < 29 - 9 \quad\Rightarrow\quad -4x < 20
2. Divide both sides by -4. Dividing by a negative reverses the sign, so < becomes >: x > \frac{20}{-4}
3. Simplify: x > -5

Answer: x > -5

Check: Try x = 0: 9 - 0 = 9 and 9 < 29 ✓. Try x = -6: 9 + 24 = 33 and 33 < 29 is false ✓. Boundary x = -5: 9 + 20 = 29, and 29 < 29 is false, so the sign stays strict.

Tip: In 9 - 4x, the minus sign belongs to the 4x. Rewrite it as -4x + 9 if that helps you see that the coefficient is -4.

Check each step:

1. Step 1: Distribute the 2 to both terms: 2 \cdot 5 - 2 \cdot x = 10 - 2x. So 10 - 2x > 4 ✓.
2. Step 2: Subtract 10 from both sides: -2x > 4 - 10 = -6 ✓.
3. Step 3: Divide both sides by -2. Because -2 is negative, the sign must flip: x < \frac{-6}{-2} \quad\Rightarrow\quad x < 3 Keisha wrote x > 3 — she forgot to flip. ✗
4. Step 4 only restates Step 3, so it is wrong because of Step 3.

Answer: Step 3

Check: Test x = 4 (in Keisha's answer): 2(5 - 4) = 2, and 2 > 4 is false. Test x = 0 (in the correct answer x < 3): 2(5) = 10 > 4 ✓. So the correct solution is x < 3.

Tip: In "find the error" items, test one number from the student's final answer in the original inequality. If it fails, trace back to the first step where a rule was broken.

1. Distribute the 4: 4x - 8 \le 4x + 3
2. Subtract 4x from both sides: -8 \le 3
3. The variable is gone. The statement -8 \le 3 is always true — it does not depend on x.
4. So every value of x makes the original inequality true.

Answer: all real numbers

Check: Try any numbers. x = 0: -8 \le 3 ✓. x = 100: 392 \le 403 ✓. x = -50: -208 \le -197 ✓. The left side is always 11 less than the right side.

Tip: When the x-terms cancel, read what is left: a true statement (like -8 \le 3) means all real numbers; a false statement (like 5 < 2) means no solution.

1. Distribute on each side: 5x - 5 - 2x < 3x + 6 - x
2. Combine like terms on each side: 3x - 5 < 2x + 6
3. Subtract 2x from both sides (keeps the x-coefficient positive): x - 5 < 6
4. Add 5 to both sides: x < 11

Answer: x < 11

Check: Boundary x = 11: left 5(10) - 22 = 28, right 3(13) - 11 = 28, equal ✓. Inside, x = 0: left -5, right 6, and -5 < 6 ✓. Outside, x = 12: left 31, right 30, and 31 < 30 is false ✓.

Tip: Simplify each side completely before moving terms across. Then move the smaller x-term so the coefficient stays positive and you never need to flip.

1. Distribute \frac{1}{2} to both terms: \frac{1}{2}x + 3 \ge 2x - 3
2. Subtract \frac{1}{2}x from both sides: 3 \ge 1.5x - 3
3. Add 3 to both sides: 6 \ge 1.5x
4. Divide by 1.5 (positive, no flip): 4 \ge x, \text{ which means } x \le 4
5. Only one choice is less than or equal to 4: 4 itself.

Answer: x = 4

Check: Plug in x = 4: left \frac{1}{2}(10) = 5, right 8 - 3 = 5, and 5 \ge 5 ✓. Plug in 4.5: left 5.25, right 6; 5.25 \ge 6 is false ✓.

Tip: In "which value is a solution" questions, plugging the choices into the original inequality is a reliable backup — even when you also solved it.

1. The fraction bar groups x + 8, so x + 8 is divided by 3. Undo the division first: multiply both sides by 3 (positive, no flip): x + 8 \le 15
2. Subtract 8 from both sides: x \le 7

Answer: x \le 7

Check: Boundary x = 7: \frac{7 + 8}{3} = \frac{15}{3} = 5 \le 5 ✓. Inside, x = 1: \frac{9}{3} = 3 \le 5 ✓. Outside, x = 10: \frac{18}{3} = 6 \le 5 is false ✓.

Tip: A fraction bar acts like parentheses around the top. When the +8 is inside the fraction, clear the denominator first; when it is outside (as in \frac{x}{3} + 8), undo the +8 first.

1. Distribute -3 to both terms: -3 \cdot x = -3x and -3 \cdot (-2) = +6. 4 - 3x + 6 > 2x - 15
2. Combine like terms on the left: 10 - 3x > 2x - 15
3. Subtract 2x from both sides: 10 - 5x > -15
4. Subtract 10 from both sides: -5x > -25
5. Divide by -5 and flip the sign: x < 5

Answer: x < 5

Check: x = 0: left 4 - 3(-2) = 10, right -15; 10 > -15 ✓. x = 6: left 4 - 12 = -8, right -3; -8 > -3 is false ✓.

Tip: Treat the minus sign in front of the parentheses as part of the multiplier: -3(x - 2) = -3x + 6.

1. Multiplying both sides by 6 must multiply every term: 6 \cdot \frac{x}{3} - 6 \cdot 4 \ge 6 \cdot \frac{x}{2} - 6 \cdot 5 \quad\Rightarrow\quad 2x - 24 \ge 3x - 30 Tomás multiplied only the fractions and left -4 and -5 unchanged, so Step 1 is wrong.
2. Steps 2–4 follow correctly from his Step 1 (including the flip in Step 4), but they start from a wrong line.
3. Correct solution: subtract 3x → -x - 24 \ge -30; add 24 → -x \ge -6; divide by -1 and flip → x \le 6.

Answer: The first error is in Step 1; the constants were not multiplied by 6. The correct solution is x \le 6.

Check: x = 6: left 2 - 4 = -2, right 3 - 5 = -2; -2 \ge -2 ✓. x = 3 (not in Tomás's answer but in the correct one): left -3, right -3.5; -3 \ge -3.5 ✓ — so x \le 1 is too small.

Tip: When you clear fractions, draw an arrow from the multiplier to every term on both sides, including whole numbers.

1. At the boundary x = -4, the two sides are equal: a(-4) + 7 = 19
2. Subtract 7: -4a = 12
3. Divide by -4: a = -3
4. Direction check: the given solution has \le while the original has \ge, so the sign was flipped — that only happens when you divide by a negative coefficient. a = -3 is negative ✓.

Answer: a = -3

Check: Solve -3x + 7 \ge 19: subtract 7 → -3x \ge 12; divide by -3 and flip → x \le -4 ✓.

Tip: In work-backwards inequality items, the boundary gives you the number and the direction tells you the sign: a flipped symbol means the coefficient was negative.

1. Multiply both sides by -2. It is negative, so flip > to <: 3x - 4 < -14
2. Add 4 to both sides: 3x < -10
3. Divide by 3 (positive, no flip): x < -\frac{10}{3} \approx -3.33
4. On a number line, -3.33 lies between -4 and -3. The numbers less than -3.33 are to its left, so the greatest integer is -4. (-3 is to the right of -3.33, so it is not a solution.)

Answer: -4

Check: x = -4: \frac{-12 - 4}{-2} = \frac{-16}{-2} = 8 > 7 ✓. x = -3: \frac{-13}{-2} = 6.5 > 7 is false ✓.

Tip: "Rounding" a negative boundary is where most errors happen. Always place the boundary on a number line and move in the shaded direction to the first integer.

1. Distribute -3 to both terms: -3 \cdot x = -3x and -3 \cdot (-4) = +12. So the first line must be -3x + 12 \ge 2x - 8. Ana is right; Ben wrote -12 — a sign error.
2. Ana's next step: subtract 2x and subtract 12 → -5x \ge -20 ✓.
3. Ana divides by -5 and flips: x \le 4 ✓. (Ben's flip is also correct, but his work started from a wrong line.)

Answer: Ana is correct; Ben made a sign error when distributing.

Check: x = 4: left -3(0) = 0, right 8 - 8 = 0; 0 \ge 0 ✓. x = 0 (in Ana's set, not in Ben's): left 12, right -8; 12 \ge -8 ✓, so Ben's answer x \le -0.8 leaves out real solutions.

Tip: A negative number times a negative number is positive. When distributing a negative, write each product separately before combining.