**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 3^{x}= 2 and \(3^{x}=\frac{2}{3}\left ( or 3^{x+1} =2\right )\)* OR*: Obtain 3^{x}= 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 3^{x}= a (or 3^{x+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 a^{b} = b log a property

Obtain –0.161 and no other answer

**Question**

**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} >,⩾, =, 2^{2}(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**

**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**

**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**

**Question**

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

**Answer/Explanation**

*Question*

Solve the equation 4|5^{x} − 1| = 5^{x}, 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(5^{x} – 1) = ± 5^{x}.

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 5^{x} = a, or 5^{x+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 2^{2} (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**

**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**

**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**

**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} > 3^{2} (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< \(-\frac{9}{5}\)