Miscellaneous Exercise answers: Linear Inequalities

Open NCERT p. 98Matches NCERT’s answer

1. The statement 2 ≤ 3x − 4 ≤ 5 means both 2 ≤ 3x − 4 and 3x − 4 ≤ 5 at the same time.
2. Add 4 to all three parts: 6 ≤ 3x ≤ 9.
3. Divide all three parts by 3 (a positive number, so the symbols do not flip): 2 ≤ x ≤ 3.

Answerx can be any real number from 2 to 3, both included.

Watch this explained “A chain turned end for end”, 11:34 into The one rule that differs from equation-solving, and the reason it differs

Open NCERT p. 98Matches NCERT’s answer

1. First expand: −3(2x − 4) = −6x + 12.
2. So the statement is 6 ≤ −6x + 12 < 12.
3. Subtract 12 from all three parts: −6 ≤ −6x < 0.
4. Divide all three parts by −6. Dividing by a negative number flips both symbols: 1 ≥ x > 0.
5. Written the usual way, this is 0 < x ≤ 1.

Answerx can be any real number greater than 0 and up to (and including) 1.

Watch this explained “A chain turned end for end”, 11:34 into The one rule that differs from equation-solving, and the reason it differs

Open NCERT p. 98Matches NCERT’s answer

1. Subtract 4 from all three parts: −7 ≤ −7x/2 ≤ 14.
2. Multiply all three parts by 2: −14 ≤ −7x ≤ 28.
3. Divide all three parts by −7. Dividing by a negative number flips both symbols: 2 ≥ x ≥ −4.
4. Written the usual way, this is −4 ≤ x ≤ 2.

Answerx can be any real number from −4 to 2, both included.

Watch this explained “A chain turned end for end”, 11:34 into The one rule that differs from equation-solving, and the reason it differs

Open NCERT p. 98Matches NCERT’s answer

1. Multiply all three parts by 5: −75 < 3(x − 2) ≤ 0.
2. Divide all three parts by 3 (positive, so no flip): −25 < x − 2 ≤ 0.
3. Add 2 to all three parts: −23 < x ≤ 2.

Answerx can be any real number greater than −23 and up to (and including) 2.

Watch this explained “A chain turned end for end”, 11:34 into The one rule that differs from equation-solving, and the reason it differs

Open NCERT p. 98Matches NCERT’s answer

1. The middle part has a minus sign in front of a fraction that already has a minus sign in its denominator, 4 − 3x/(−5). Those two minus signs cancel: 3x/(−5) = −3x/5, so 4 − (−3x/5) = 4 + 3x/5.
2. Because the two minus signs cancel, the coefficient of x here is positive, so no symbol needs to flip on that account.
3. So the statement is −12 < 4 + 3x/5 ≤ 2.
4. Subtract 4 from all three parts: −16 < 3x/5 ≤ −2.
5. Multiply all three parts by 5/3 (positive, so no flip): −80/3 < x ≤ −10/3.

Answerx can be any real number greater than −80/3 and up to (and including) −10/3.

Watch this explained “A minus that is not a multiplier”, 12:51 into The one rule that differs from equation-solving, and the reason it differs

Open NCERT p. 98Matches NCERT’s answer

1. Multiply all three parts by 2: 14 ≤ 3x + 11 ≤ 22.
2. Subtract 11 from all three parts: 3 ≤ 3x ≤ 11.
3. Divide all three parts by 3 (positive, so no flip): 1 ≤ x ≤ 11/3.

Answerx can be any real number from 1 to 11/3, both included.

Watch this explained “A chain turned end for end”, 11:34 into The one rule that differs from equation-solving, and the reason it differs

Open NCERT p. 98Matches NCERT’s answer

1. Solve the first inequality on its own: 5x + 1 > −24, so 5x > −25, so x > −5.
2. Solve the second inequality on its own: 5x − 1 < 24, so 5x < 25, so x < 5.
3. Both must hold together, so x must lie in both answers: x > −5 and x < 5.
4. The lower end −5 comes from the first inequality; the upper end 5 comes from the second.
5. On the number line, draw a hollow circle at −5 and a hollow circle at 5 (neither end is included) and shade the part of the line between them.

Answerx can be any real number between −5 and 5, not including either end: −5 < x < 5.

Watch this explained “The shapes an overlap comes in”, 11:08 into Drawing the answer as a piece of the number line, hollow circle or solid

Open NCERT p. 98Matches NCERT’s answer

1. Solve the first inequality: 2(x − 1) < x + 5, so 2x − 2 < x + 5, so x < 7.
2. Solve the second inequality: 3(x + 2) > 2 − x, so 3x + 6 > 2 − x, so 4x > −4, so x > −1.
3. Both must hold together: x > −1 and x < 7.
4. The lower end −1 comes from the second inequality; the upper end 7 comes from the first.
5. On the number line, draw a hollow circle at −1 and a hollow circle at 7 (neither end is included) and shade the part of the line between them.

Answerx can be any real number between −1 and 7, not including either end: −1 < x < 7.

Watch this explained “The shapes an overlap comes in”, 11:08 into Drawing the answer as a piece of the number line, hollow circle or solid

Open NCERT p. 98Matches NCERT’s answer

1. Solve the first inequality: 3x − 7 > 2(x − 6), so 3x − 7 > 2x − 12, so x > −5.
2. Solve the second inequality: 6 − x > 11 − 2x, so x > 5.
3. Both must hold together. Every number greater than 5 is already greater than −5, so the second condition decides everything.
4. The first condition adds nothing extra here.
5. On the number line, draw a hollow circle at 5 (5 itself is not included) and shade the line to the right of it, with an arrow to show it goes on without end.

Answerx can be any real number greater than 5: x > 5.

Watch this explained “The shapes an overlap comes in”, 11:08 into Drawing the answer as a piece of the number line, hollow circle or solid

Open NCERT p. 98Matches NCERT’s answer

1. Solve the first inequality: 5(2x − 7) − 3(2x + 3) ≤ 0, so 10x − 35 − 6x − 9 ≤ 0, so 4x ≤ 44, so x ≤ 11.
2. Solve the second inequality: 2x + 19 ≤ 6x + 47, so −28 ≤ 4x, so x ≥ −7.
3. Both must hold together: −7 ≤ x ≤ 11.
4. The lower end −7 comes from the second inequality; the upper end 11 comes from the first.
5. On the number line, draw a filled (solid) circle at −7 and a filled circle at 11 (both ends are included) and shade the part of the line between them.

Answerx can be any real number from −7 to 11, both included: −7 ≤ x ≤ 11.

Watch this explained “Two ends, two reasons”, 10:00 into Drawing the answer as a piece of the number line, hollow circle or solid

Open NCERT p. 98Checked by computer

1. The temperature is kept between 68° F and 77° F, so 68 < F < 77.
2. Substitute F = (9/5)C + 32: 68 < (9/5)C + 32 < 77.
3. Subtract 32 from all three parts: 36 < (9/5)C < 45.
4. Multiply all three parts by 5/9 (positive, so no flip): 20 < C < 25.

AnswerThe temperature is between 20°C and 25°C (not including either end).

Watch this explained “A chain turned end for end”, 11:34 into The one rule that differs from equation-solving, and the reason it differs

Open NCERT p. 98Matches NCERT’s answer

1. Let x litres of the 2% solution be added.
2. Pure boric acid in the mixture = 8% of 640 + 2% of x = 51.2 + 0.02x litres.
3. Total mixture = (640 + x) litres.
4. The mixture must be more than 4% and less than 6% boric acid: 4% of (640+x) < 51.2 + 0.02x < 6% of (640+x).
5. The left part gives 25.6 + 0.04x < 51.2 + 0.02x, so 0.02x < 25.6, so x < 1280.
6. The right part gives 51.2 + 0.02x < 38.4 + 0.06x, so 12.8 < 0.04x, so x > 320.

AnswerMore than 320 litres but less than 1280 litres of the 2% solution must be added.

Watch this explained “Two constraints at once”, 9:10 into Drawing the answer as a piece of the number line, hollow circle or solid

Open NCERT p. 99Matches NCERT’s answer

1. Let x litres of water be added. Water has 0% acid, so the amount of acid stays fixed.
2. Pure acid in the mixture = 45% of 1125 = 506.25 litres.
3. Total mixture = (1125 + x) litres.
4. The mixture must contain more than 25% and less than 30% acid: 25% of (1125+x) < 506.25 < 30% of (1125+x).
5. The left part gives 281.25 + 0.25x < 506.25, so 0.25x < 225, so x < 900.
6. The right part gives 506.25 < 337.5 + 0.3x, so 168.75 < 0.3x, so x > 562.5.

AnswerMore than 562.5 litres but less than 900 litres of water must be added.

Watch this explained “Two constraints at once”, 9:10 into Drawing the answer as a piece of the number line, hollow circle or solid

Open NCERT p. 99Matches NCERT’s answer

1. Here CA = 12, so IQ = (MA/12) × 100.
2. We are given 80 ≤ IQ ≤ 140, so 80 ≤ (MA/12) × 100 ≤ 140.
3. Divide all three parts by 100: 0.8 ≤ MA/12 ≤ 1.4.
4. Multiply all three parts by 12 (positive, so no flip): 9.6 ≤ MA ≤ 16.8.

AnswerThe mental age lies between 9.6 and 16.8 years, both included.

Watch this explained “A chain turned end for end”, 11:34 into The one rule that differs from equation-solving, and the reason it differs