Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter 8
Worked example (no skipped steps)
8.1 Solve 𝑥 + 4 > 9.
Subtract 4 from both sides:
𝑥 + 4 − 4 > 9 − 4
𝑥 > 5
Answer: 𝑥 > 5.
Worked example (no skipped steps)
8.2 Solve 𝑦 − 3 ≤ 7.
Add 3 to both sides:
𝑦 − 3 + 3 ≤ 7 + 3
𝑦 ≤ 10
Answer: 𝑦 ≤ 10.
Worked example (no skipped steps)
8.3 Solve 2 𝑥 < 10.
Divide both sides by 2:
2 𝑥
10
<
2
2
𝑥 < 5
Answer: 𝑥 < 5.
167
Worked example (no skipped steps)
8.4 Solve −3 𝑥 ≤ 12.
Divide both sides by −3. Because we divide by a negative number, reverse the inequality: 𝑥 ≥ −4
Answer: 𝑥 ≥ −4.
Worked example (no skipped steps)
8.5 Solve 5 − 2 𝑘 ≥ 1.
Subtract 5 from both sides:
−2 𝑘 ≥ −4
Divide by −2 and reverse the sign:
𝑘 ≤ 2
Answer: 𝑘 ≤ 2.
Worked example (no skipped steps)
8.6 Solve 2 < 𝑥 + 1 ≤ 6.
Subtract 1 from all three parts:
2 − 1 < 𝑥 + 1 − 1 ≤ 6 − 1
1 < 𝑥 ≤ 5
Answer: 1 < 𝑥 ≤ 5.
Worked example (no skipped steps)
8.7 Write the solution to 𝑥 ≥ −2 in interval notation.
The value −2 is included, so use a bracket at −2. The solution goes to the right forever:
[−2 , ∞)
Answer: [−2 , ∞).
Worked example (no skipped steps)
8.8 Write the solution to 𝑥 < 4 in interval notation.
The value 4 is not included, so use a parenthesis at 4. The solution goes left forever: (−∞ , 4)
Answer: (−∞ , 4).
Step-by-step solutions
Worked example (no skipped steps)
8.9 Solve 𝑥 + 4 > 11.
Subtract 4 from both sides:
𝑥 > 7
Answer: 𝑥 > 7.
Worked example (no skipped steps)
8.10 Solve 3 𝑥 − 5 ≤ 7.
Add 5 to both sides:
3 𝑥 ≤ 12
Divide by 3:
𝑥 ≤ 4
Answer: 𝑥 ≤ 4.
Worked example (no skipped steps)
8.11 Solve −2 𝑥 < 10.
Divide by −2 and reverse the sign:
𝑥 > −5
Answer: 𝑥 > −5.
Worked example (no skipped steps)
8.12 Solve 5 − 3 𝑥 ≥ −7.
Subtract 5 from both sides:
−3 𝑥 ≥ −12
Divide by −3 and reverse the sign:
𝑥 ≤ 4
Answer: 𝑥 ≤ 4.
Worked example (no skipped steps)
8.13 Solve −3 ≤ 𝑥 + 2 < 9.
Subtract 2 from all three parts:
−3 − 2 ≤ 𝑥 + 2 − 2 < 9 − 2
−5 ≤ 𝑥 < 7
Answer: −5 ≤ 𝑥 < 7.
169
Worked example (no skipped steps)
8.14 Solve −4 ≤ 2 𝑦 − 2 < 8.
Add 2 to all three parts:
−2 ≤ 2 𝑦 < 10
Now divide all three parts by 2:
−1 ≤ 𝑦 < 5
Answer: −1 ≤ 𝑦 < 5.
Worked example (no skipped steps)
8.15 Solve 7 − 2 𝑚 > 1.
Subtract 7:
−2 𝑚 > −6
Divide by −2 and reverse the sign:
𝑚 < 3
Answer: 𝑚 < 3.
Worked example (no skipped steps)
8.16 Solve 𝑥 − 4 ≥ −1.
3
Add 4 to both sides:
𝑥 ≥ 3
3
Multiply both sides by 3:
𝑥 ≥ 9
Answer: 𝑥 ≥ 9.