Inequalities Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Inequalities questions. See exactly how to solve problems on modulus, modulus-inequality, critical-values, quadratic-inequality.

modulusmodulus-inequalitycritical-valuesquadratic-inequalitygraphical-methodrational-inequality
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
Solve the inequality ∣2x−1∣<5\left|{2x-1}\right|<5.

Worked solution

  1. Remove the modulus using the double-inequality rule

    −5  <  2x−1  <  5-5\;<\;2x-1\;<\;5

    For a positive constant kk, ∣u∣<k\left|u\right|<k is equivalent to −k<u<k-k<u<k.

  2. Isolate the term in xx in all three parts

    −4  <  2x  <  6-4\;<\;2x\;<\;6

    The constant −1-1 is removed from all three parts at once.

  3. Divide all three parts by 22

    −2  <  x  <  3-2\;<\;x\;<\;3

    Dividing by a positive number does not reverse the inequality.

  4. State the complete solution set

    −2<x<3-2<x<3

    This is the complete set of values of xx satisfying the inequality.

Answer
−2<x<3-2<x<3
Question 2
2 markseasy
Find the complete set of values of xx for which ∣x+1∣<4\left|{x+1}\right|<4.

Worked solution

  1. Remove the modulus using the double-inequality rule

    −4  <  x+1  <  4-4\;<\;x+1\;<\;4

    For a positive constant kk, ∣u∣<k\left|u\right|<k is equivalent to −k<u<k-k<u<k.

  2. Isolate the term in xx in all three parts

    −5  <  x  <  3-5\;<\;x\;<\;3

    The constant 11 is removed from all three parts at once.

  3. State the critical values

    x=−5,x=3x=-5,\quad x=3

    These are the values at which the two sides are equal or the expression is undefined.

  4. State the complete solution set

    −5<x<3-5<x<3

    This is the complete set of values of xx satisfying the inequality.

Answer
−5<x<3-5<x<3
Question 3
2 markseasy
Solve ∣3x+2∣≤8\left|{3x+2}\right|\le 8, giving your answer in inequality notation.

Worked solution

  1. Remove the modulus using the double-inequality rule

    −8  ≤  3x+2  ≤  8-8\;\le\;3x+2\;\le\;8

    For a positive constant kk, ∣u∣<k\left|u\right|<k is equivalent to −k<u<k-k<u<k.

  2. Isolate the term in xx in all three parts

    −10  ≤  3x  ≤  6-10\;\le\;3x\;\le\;6

    The constant 22 is removed from all three parts at once.

  3. Divide all three parts by 33

    −103  ≤  x  ≤  2-\frac{10}{3}\;\le\;x\;\le\;2

    Dividing by a positive number does not reverse the inequality.

  4. State the complete solution set

    −103≤x≤2-\frac{10}{3}\le x\le 2

    This is the complete set of values of xx satisfying the inequality.

Answer
−103≤x≤2-\frac{10}{3}\le x\le 2
Question 4
2 markseasy
Solve the inequality ∣x−4∣>3\left|{x-4}\right|>3.

Worked solution

  1. Split the modulus into two separate cases

    x−4  <  −3orx−4  >  3x-4\;<\;-3\quad\text{or}\quad x-4\;>\;3

    For a positive constant kk, ∣u∣>k\left|u\right|>k is equivalent to u<−ku<-k or u>ku>k.

  2. Solve the first case

    x−4<−3  ⇒  x<1x-4<-3\;\Rightarrow\;x<1

    Rearranging the first branch gives one part of the solution.

  3. Solve the second case

    x−4>3  ⇒  x>7x-4>3\;\Rightarrow\;x>7

    Rearranging the second branch gives the other part.

  4. State the complete solution set

    x<1orx>7x<1\quad\text{or}\quad x>7

    This is the complete set of values of xx satisfying the inequality.

Answer
x<1orx>7x<1\quad\text{or}\quad x>7
Question 5
2 markseasy
Solve the inequality ∣2x+5∣≥9\left|{2x+5}\right|\ge 9, stating your critical values clearly.

Worked solution

  1. Split the modulus into two separate cases

    2x+5  ≤  −9or2x+5  ≥  92x+5\;\le\;-9\quad\text{or}\quad 2x+5\;\ge\;9

    For a positive constant kk, ∣u∣>k\left|u\right|>k is equivalent to u<−ku<-k or u>ku>k.

  2. Solve the first case

    2x+5≤−9  ⇒  x≤−72x+5\le -9\;\Rightarrow\;x\le -7

    Rearranging the first branch gives one part of the solution.

  3. Solve the second case

    2x+5≥9  ⇒  x≥22x+5\ge 9\;\Rightarrow\;x\ge 2

    Rearranging the second branch gives the other part.

  4. State the complete solution set

    x≤−7orx≥2x\le -7\quad\text{or}\quad x\ge 2

    This is the complete set of values of xx satisfying the inequality.

Answer
x≤−7orx≥2x\le -7\quad\text{or}\quad x\ge 2

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