## Math 8 Chapter 4 Lesson 5: Equations containing absolute sign

## 1. Theoretical Summary

### 1.1. Again about absolute value

The absolute value of the number \(a\), denoted by \(|a|\) is defined as follows:

\(|a| = a\) when \(a ≥ 0\)

\(|a| = -a\) when \(a < 0\)

### 1.2. Solve some equations containing absolute sign

- Step 1: Apply the absolute value definition to remove the absolute value sign
- Step 2: Solve equations with no absolute sign
- Step 3: Choose the appropriate solution in each case under consideration
- Step 4: Conclusion of the test.

## 2. Illustrated exercise

### 2.1. Exercise 1

Simplify expressions:

a) \(C = |-3x| + 7x – 4\) when \(x ≤ 0\);

b) \(D = 5 – 4x + |x – 6|\) when \(x < 6\).

**Solution guide**

a) \(x ≤ 0\) so \(– 3x ≥ 0 ⇒ |-3x| = -3x\)

So \(C = |-3x| + 7x – 4 \)\(\,= -3x + 7x – 4 = 4x – 4\)

b) \(x < 6\) so \(x – 6 < 0\) \(⇒ |x - 6| = -(x - 6) = 6 - x\)

So \(D = 5 – 4x + |x – 6| \)\(\,= 5 – 4x + 6 – x = 11 – 5x\).

### 2.2. Exercise 2

Solve the equations:

a) \(|x + 5| = 3x + 1\);

b) \(|-5x| = 2x + 21\).

**Solution guide**

a) With \(x ≥ -5\) then \(x + 5 ≥ 0\) so \(|x + 5| = x + 5\)

Then: \(|x + 5| = 3x + 1\)

\(\Rightarrow x + 5 = 3x + 1 \)

\( \Leftrightarrow x – 3x = 1 – 5\)

\(⇔ -2x = -4 \)

\( \Leftrightarrow x = \left( { – 4} \right):\left( { – 2} \right)\)

\(⇔ x = 2\) (satisfy the condition \(x ≥ -5\))

With \(x < -5\) then \(x + 5 < 0\) so \(|x + 5| = - (x + 5) = - x - 5\)

Then: \(|x + 5| = 3x + 1\)

\( \Rightarrow -x – 5 = 3x + 1\)

\( \Leftrightarrow – x – 3x = 1 + 5\)

\(⇔ -4x = 6 \)

\( \Leftrightarrow x = 6:\left( { – 4} \right)\)

\(⇔ x = \dfrac{{ – 3}}{2}\) (does not satisfy the condition \(x < -5\))

So the solution set of the equation \(|x + 5| = 3x + 1\) is \(S = \{2\}\)

b) With \(x ≥ 0\) then \(- 5x ≤ 0\) so \(|-5x| = -(-5x) = 5x\)

Then: \(|-5x|= 2x + 21\)

Deduce \( 5x = 2x + 21\)

\( \Leftrightarrow 5x – 2x = 21\)

\(⇔ 3x = 21 \)

\( \Leftrightarrow x = 21:3\)

\(⇔ x = 7\) (satisfy the condition \(x ≥0\))

With \(x < 0\) then \(– 5x > 0\) so \(|-5x| = -5x\)

Then: \(|-5x|= 2x + 21 \)

So \( -5x = 2x + 21\)

\( \Leftrightarrow – 5x – 2x = 21\)

\(⇔ -7x = 21\)

\( \Leftrightarrow x = 21:\left( { – 7} \right)\)

\(⇔ x = -3\) (satisfy the condition \(x < 0\))

So the solution set of the equation \(|-5x|= 2x + 21\) is \(S = \{7;-3\}\)

### 2.3. Exercise 3

Unsign the absolute value and reduce the expression: \(A = 3x + 2 + |5x| \) in two cases: \(x ≥ 0\) and \(x < 0\)

**Solution guide**

\(A = 3x + 2 + |5x| \)

– When \(x ≥ 0\) we have \(5x ≥ 0\) so \(|5x| =5x\).

Hence \(A = 3x + 2 + 5x = 8x + 2 \) when \(x ≥ 0\).

– When \(x < 0\) we have \(5x < 0\) so \(|5x| = -5x\).

Hence \(A = 3x + 2 – 5x = -2x + 2 \) when \(x < 0\).

So \(A = 8x + 2 \) when \(x ≥ 0\);

\(A = -2x + 2\) when \(x < 0\).

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Solve the equations

a) \(\left| {0,5x} \right| = 3 – 2x\)

b) \(\left| { – 2x} \right| = 3x + 4\)

c) \(\left| {5x} \right| = x – 12\)

**Verse 2:** Solve the equations

a) \(\left| {9 + x} \right| = 2x\)

b) \(\left| {x – 1} \right| = 3x + 2\)

c) \(\left| {x + 6} \right| = 2x + 9\)

**Question 3:** Solve the equations:

a) \(\left| {5x} \right| – 3x – 2 = 0;\)

b) \(x – 5x + \left| { – 2x} \right| – 3 = 0;\)

c) \(\left| {3 – x} \right| + {x^2} – \left( {4 + x} \right)x = 0;\)

**Question 4:** Solve the equations:

a) \(\left| {x – 5} \right| = 3\)

b) \(\left| {x + 6} \right| = 1\)

c) \(\left| {2x – 5} \right| = 4\)

### 3.2. Multiple choice exercises

**Question 1:** Which of the following is incorrect:

A. \(|\pm 1|=1\)

B. \(|-x^{2}|=x^{2}\)

C. \(|x|=x\)

D. \(|x^{2}+2|=x^{2}+2\)

**Verse 2: **Which of the following is incorrect:

A. |1.75|=1.75

B. \(|\pm \frac{3}{4}|=\pm \frac{3}{4}\)

C. \(|-\sqrt{3}|= \sqrt{3}\)

D. |0|=0

**Question 3:** Choose the correct answer

Given Q=|-3x|+2x when \(x \neq 0\) then:

A. Q=-5+x

B. Q=-1+2x

C. Q=5x

D. Q=-x

**Question 4: **Choose the correct answer.

Given K=|-2004x|+203x when x<0 then:

A. K=x

B. K=-4007x

C. K=4006x

D. K=-x

**Question 5: **The solution set of the equation |x-8|=2x+5 is:

A. {-1}

B. {1;-13}

C. {1}

D. {1;13}

## 4. Conclusion

Through this lesson, you will learn some of the main topics as follows:

- Recognize how to unsign absolute values in expressions of form and form
- Solve some equations of the form = cx + d and of the form = cx + d.

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