Inequality is a mathematical statement that compares two expressions. The statement resembles the usual equation except that the equality sign (=) is replaced by one of the following inequality signs

< is less than

> is greater than

≤ is less than or equal to

≥ is greater than or equal to

Solving inequalities does not create any problem, as the approach, manner and steps are the same as in linear equations. The only slight difference is that equality sign is changed to the appropriate inequality sign in the solution.

It is important for us to note that an inequality remains true if:

- We add the same number to both sides
- We subtract the same number from both sides of the inequality
- We multiply or divide both sides of the inequality by same positive number
- We multiply or divide both sides of the inequality by same negative number if and only if we change the inequality sign i.e> to <, <to>, ≤ to ≥, ≥ to ≤

Note: solutions of inequalities are always a range of values.

With the facts we have established on this topic, we should proceed in solving some challenging problems.

**Example 1:**

Solve the inequality and show the solution set in number line

*2x+5/x *≤ *2x/x-3*

**Solution**

*2x+5/x *≤ *2x/x-3*

Multiply both sides of the equation by the L.C.M *x(x-3)*

*X(x-3) 2x+5/x ≤ 2x(x-3)/x-3*

*(x-3) (2x+5) ≤ 2x(x)*

Open the bracket

*x *x*2x + *xx*5 + (-3*x*2x) + (-*3x*5)*

*≤ 2x ^{2}*

Collect like terms

*2x ^{2} + 5x – 6x – 15 ≤ 2x^{2}*

*2x – 2x ^{2} = 0*

*5x – 6x – 15 ≤ 0*

*-x – 15 ≤ 0*

Take *-15 *to the right hand side

*-x ≤ 15*

Divide both sides by -1

*-1x/-1 ≤ +5/-1*

*x ≥ -15*

N/B: we stated earlier, when you multiply or divide inequality by a negative number, the sign changes.

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**Example 2:**

Solve the inequality

*5*x *-1/3 –x/2 ≤ 8+x*

Solution

*5 *x *-1/3 –x/2 ≤ 8+x*

The L.C.M of 3 and 2 is multiply each side of the equation by 6 to clear the fraction

*6(5*x*-1)/3 = 6*x*x/2 ≤ (8+x)6*

*2(5x-1)-3x-48+6x*

*10x – 2 – 3x ≤ 48 + 6x*

Collect like terms

*10x – 2 – 3x ≤ 48 + 6x*

*10x – 6x – 3x ≤ 48 + 2*

*10x – 9x ≤ 50*

*x ≤50*

**Example 3:**

Solve the inequality

½ (*4x-7) – 1/3(1-4x) ≥ 6*

Solution

The equation becomes

(*4x-7)/2 – (1-4x)/3 7,6*

Multiply each side of the equation 6 to clear the fraction

*6(4x-7)/2 – 6(1-4x)/2 ≥ 6 *x*6*

*3(4x-7) -2 (1-4x) ≥ 36*

*12x – 21 – 2 + 8x ≥ 36*

Collect like terms

*12x + 8x ≥ 36 + 21 + 2*

*20x/20 ≥ 59/20*

*x ≥ 2 19/20*

**Example 4:**

Solve the following inequalities and show the solution set in number line

*4x – 1 ≤ 7**3x-5 < 5x – 3**2/5*x*-7 >2x – 9*

Solution

*4x – 1 ≤ 7*

Collect like terms

*4x ≤ 7+1*

*4x/4 ≤ 8/4*

*X ≤ 8/4*

NUMBER LINE

**Example 4:**

*3x-5 ≤ 5x -3*

Collect like terms

*3x-5x ≤ – 3*

*-2 x ≤ -3 + 5*

Divide both sides by -2

*-2x/-2 ≤ +2/+2 – 2/2*

*x ≥ -1*

NUMBER LINE

Since we divide both sides by a negative number, the sign changes

*2/5x -7>2x-9*

Solution

*2/5x -7 >2x-9*

Multiply both sides of the equation by 5

*5(2x)/5 – 7 *x* 5 >2x *x* 5 – 9*x*5*

*10x/4 – 35 >10x – 45*

*2x – 35 > 10x – 45*

*2x – 35 > 10x – 45*

Collect like terms

*2x-10x > – 45 + 35*

*-8x/-8 > +10/+8*

*x< 10/8*

*x< 5/4*

*x< 1.25*

the sign changes because if divided it with a negative number

NUMBER LINE

**Example 6**

Solve the inequality 1*/3 x + 1 ≤ 2x/3x – 5*

Solution

Multiply both sides of the equation by 3 into year the fraction

*3*x*x+/3/3 + 3×1 ≤ 3×2/3*x – 3 x 5

3s/3 + 3 *≤ 6/3 – 15*

*x + 3 ≤ 2x – 15*

collect like terms

*x – 2x ≤ – 15 = 3*

*-x – 2x -18*

Divide both sides by -1

-1/+1 *≤ -18/-1*

*x ≥ 18*

the sign changes because we divide both sides by -1

Your ability to know the inequality signs and use them effectively determines your capability to solve inequality problems.

When you multiply or divide inequality with a negative number, follow the rule involved and you won’t have any challenge when solving the inequality equation.