Question

Find the set of values of x satisfying the inequality 2|2x − a |< |x + 3a |, where a is a positive constant.

Answer/Explanation                                                   

Question

Solve the inequality| 2x − 3 |> 4|x + 1|.

State or imply non-modular inequality\( (2x-3)^{2}>4^{2}(x+1)\)

Answer/Explanation

(a^{2}\),or corresponding quadratic equation, or pair of linear equations \( (2x-3)=\pm 4(x-1)\)

Make reasonable attempt at solving a 3-term quadratic, or solve two linear equations
for x

Obtain critical values x=-\(\frac{7}{2}and x= -\frac{1}{6}\)

State final answer -\(\frac{7}{2}<x<-\frac{1}{6}\)

Alternative method for question

Obtain critical value x = -\(\frac{7}{2}\)  from a graphical method, or by inspection, or by solving a linear equation or an inequality 

Obtain critical value x=-\(\frac{1}{6}\) similarly
State final answer -\(\frac{7}{2}<x<-\frac{1}{6}\)

Question

 Solve the equation \(2\left | 3^{x}-1 \right |=3^{x}\) , giving your answers correct to 3 significant figures. [4]

Answer/Explanation

Ans:

 EITHER: State or imply non-modular equation \(2^{2}\left ( 3^{x}-1 \right )^{2}=\left ( 3^{x} \right )^{2}\) , or pair of equations \(2\left ( 3^{x}-1 \right )= \pm 3^{x}\)
                      Obtain 3x= 2 and \(3^{x}=\frac{2}{3}\left ( or 3^{x+1} =2\right )\)
    OR:           Obtain 3x= 2 by solving an equation or by inspection 
                      Obtain \(3^{x}=\frac{2}{3}\left ( or 3^{x+1} =2\right )\) by solving an equation or by inspection 

Use correct method for solving an equation of the form 3x= a (or 3x+1= a), where a > 0

Obtain final answers 0.631 and –0.369 

Question

 (i) Solve the equation \(\left | 4x-1 \right |=\left | x-3 \right |.\) [3]

   (ii) Hence solve the equation \(\left | 4^{y+1}-1 \right |=\left | 4^{y}-3 \right |\) correct to 3 significant figures. [3]

Answer/Explanation

Ans:

(i) Either State or imply non-modular equation (4x – 1)2 = (x – 3)2 or pair of linear equations 4x – 1 = ± x – 3 
                       Solve a three-term quadratic equation or two linear equations 
                       Obtain \(-\frac{2}{3}\) and \(\frac{4}{5}\)
          Or        Obtain value \(-\frac{2}{3}\) from inspection or solving linear equation
                       Obtain value \(\frac{4}{5}\) similarly

   (ii) State or imply at least \(4^{y}=\frac{4}{5}\) , following a positive answer from part (i) 
           Apply logarithms and use log ab = b log a property
           Obtain –0.161 and no other answer 

Question

Solve the inequality |2x + 1| <| 3x − 2|.

Answer/Explanation

EITHER:
State or imply non-modular inequality \((2x+1)^{2}<(3(x-2))^{2}\) or corresponding
quadratic equation, or pair of linear equations \((2x+1)\pm 3(x-2)\)

Make reasonable solution attempt at a 3-term quadratic e.g. 5x^{2}-40x+35=0 or solve two linear equations for x

Obtain critical values x = 1 and x = 7 
State final answer x < 1 and x > 7 
OR:
Obtain critical value x = 7 from a graphical method, or by inspection, or by solving
a linear equation or inequality

Obtain critical value x = 1 similarly
State final answer x < 1 and x > 7

Question

Solve the inequality 2 − 5x > 2|x − 3|.

Answer/Explanation

  Make a recognisable sketch graph of y x = 2 |x −  3| and the line
    y = 2 – 5x

    Find x-coordinate of intersection with y = 2 – 5x

    Obtain \(x=-\frac{4}{3}\)

    State final answer \(x=< -\frac{4}{3}\)

Alternative method for question 1

   State or imply non-modular inequality/equality
   (2 – 5x)2 >,⩾, =, 22(x – 3)2, or corresponding quadratic equation,
    or pair of linear equations
   (2 – 5x) >,⩾ , =, ± 2(x – 3)

    Make reasonable attempt at solving a 3-term quadratic, or solve
    one linear equation, or linear inequality for x

    Obtain critical value \(x= -\frac{4}{3}\)

    State final answer \(x< -\frac{4}{3}\)

Question

(a) Sketch the graph of y = |x-2|
(b) Solve the inequality |x-2|<3x-4.

Answer/Explanation

Ans:

(a) Make a recognisable sketch graph of y = |x-2|
(b) Find x-coordinate of intersection with y = 3x – 4
Obtain \(x=\frac{3}{2}\)
State final asnwer \(x>\frac{3}{2}\) only
Alternative method for question 1(b)
Solve the linear inequality 3x – 4 > 2 – x, or corresponding equation
Obtain critical value \(x=\frac{3}{2}\)
State final asnwer \(x>\frac{3}{2}\) only
Alternative method for question 1(b)
Solve the quadratic inequality \((x-2)^2<(3x-4)^2\).
or corresponding equation
Obtain critical value \(x=\frac{3}{2}\)
State final answer \(x>\frac{3}{2}\) only

Question

Solve the equation \((1+2^{x})=2\), giving your answer correct to 3 decimal places.

Answer/Explanation

Remove logarithm and obtain \(1+2^{x}=e^{2}\)

Use correct method to solve an equation of the form \(2^{x}\) = a , where a > 0Obtain answer x = 2.676

Question

Solve the inequality| x − 4| < 2|3x + 1|.

Answer/Explanation

Question

Solve the equation 4|5x − 1| = 5x, giving your answers correct to 3 decimal places.

Answer/Explanation

Ans:

State or imply non-modular equation \(4^{2}\left ( 5^{x} – 1\right )^{2} = \left ( 5^{x} \right )^{2}\) or pair of equations 4(5x – 1) = ± 5x.

Obtain \(5^{x} = \frac{4}{3} and 5^{x} = \frac{4}{5} (or 5^{X+1} = 4)\)

Use correct method for solving an equation of the form 5x = a, or 5x+1 = b where a> 0, or b> 0

Obtain answer x = 0.179 and x = -0.139

Question

Solve the inequality 2|3x − 1|< |x + 1|.                                                                                 [4]

Answer/Explanation

Ans

State or imply non-modular inequality  22 (3x –  1) <(x + 1) , or
    corresponding quadratic equation, or pair of linear equations

    Form and solve a 3-term quadratic, or solve two linear equations for x

    Obtain critical values \(x=\frac{3}{5}\ and \ x=\frac{1}{7}\)

    State final answer  \(\frac{1}{7}< x< \frac{3}{5}\)

    Alternative method for Question 1

    Obtain critical value  \(x-\frac{3}{5}\)  a graphical method, or by solving      a linear  equation or linear inequality

    Obtain critical value \(x-\frac{1}{7}\)  similarly

    State final answer  \(\frac{1}{7}< x< \frac{3}{5}\)

Question

(i) Solve the equation |2x − 1| = 3x.

(ii) Hence solve the equation \(|2|5^{x}-1|=3|5^{x}|\),giving your answer correct to 3 significant figures.

Answer/Explanation

(i) EITHER: State or imply non-modular equation \((2(x-1))^{2}=(3x)^{2} \), or pair of linear equations
2( 1) 3 x − =± x

Make reasonable solution attempt at a 3-term quadratic, or solve two linear equations

Obtain answers x = −2 and \(x=\frac{2}{5}\)

OR: Obtain answer x = −2 by inspection or by solving a linear equation

Obtain answer similarly \(x=\frac{2}{5}\)

(ii) Use correct method for solving an equation of the form\( 5^{x}=a or 5^{x+1}\) =a + = , where a > 0 Obtain answer x =– 0.569 only

Question

Solve the equation \( |4-2^{x}|\) =10, giving your answer correct to 3 significant figures.

Answer/Explanation

State or imply \(4-2^{x}=\)-10 and 10

Use correct method for solving equation of form \(2^{x}\)=a

Obtain 3.81

Question

Find the set of values of x satisfying the inequality 3|x − 1| < |2x + 1|.

EITHER State or imply non-modular inequality\( (3(x – 1))^{2} < (2x + 1)^{2}\)
or corresponding quadratic equation, or pair of linear equations 3(x – 1) = ± (2x + 1)

Answer/Explanation

Make reasonable solution attempt at a 3-term quadratic, or solve two linear
equations 
Obtain critical values \(x\frac{2}{5}\) and x=4

state answer\( \frac{2}{5}\)<x<4

OR  Obtain critical value x \(\frac{2}{5} \) or x=4 from a graphical method, or by inspection, or by solving a linear equation or inequality

Obtain critical values x \(\frac{2}{5} \) or x=4

State answer \(\frac{2}{5}\)<x<4

[Do not condone for .]

Question

Solve the inequality |2x + 3| > 3|x + 2|.

Answer/Explanation

Ans:

State or imply non-modular inequality (2x + 3)2 > 32 (x + 2)2 , or corresponding quadratic equation, or pair of linear equations

Make a reasonable attempt at solving a 3-term quadratic, or solve two linear equations for x

Obtain critical values x = – 3 and x = \(-\frac{9}{5}\)

State final answer -3 < x < \(-\frac{9}{5}\) or x > -3 and x< \(-\frac{9}{5}\)

Alternative method for question 1

Obtain critical value x = – 3 from a graphical method, or by solving a linear equation or linear inequality

Obtain critical value x = \(-\frac{9}{5}\) similarly

State final answer -3 < x < \(-\frac{9}{5}\) or x> -3 and x< \(-\frac{9}{5}\)