Question

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

Question

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

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

(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]

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]

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|.

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|.

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.

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.

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|.

Question

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

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]

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.

(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.

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)

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|.

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< $$-\frac{9}{5}$$