Question

Solve each system of inequalities.
|2x+5| + |x - 3|<10
1.5x-3<|x-1|

Answer 1

the solution for this inequality is:
x < 4 and × < 8/5
Overall, the system of inequalities can be represented by the solutions:
× <2, x>-8, × <4, and × < 8/5.

Answer 2

1st one: x < 2.67 or x > -4

2nd one: x < 4 or x > 1.6

Explanation:

Definition of an inequality:

An inequality is a mathematical statement that compares two expressions and shows that they are not equal. Inequalities can be written in a variety of ways, but the most common symbols are <, >, ≤, and ≥.

• The symbol < means "less than."
• The symbol > means "greater than."
• The symbol ≤ means "less than or equal to."
• The symbol ≥ means "greater than or equal to."

Solution:

In order to solve the system of inequalities, we can break it down into two separate inequalities and solve each one individually.

Inequality 1:

|2x+5| + |x - 3|<10

To solve this inequality, we can use the following steps:

Split the inequality into two cases, one for when each absolute value is positive and one for when each absolute value is negative.

Solve each case individually.

Combine the solutions from each case to get the final

Solution.

Case 1:

2x+5 + x - 3 < 10

Combining like terms, we get:

3x + 2 < 10

Subtracting 2 from both sides, we get:

3x < 8

Dividing both sides by 3, we get:

`\sf (3x )/(3) <( 8)/(3)`

x < 2.67

Case 2:

-(2x+5) -(x - 3) < 10

Distributing the negative sign, we get:

-2x - 5 -x + 3 < 10

Combining like terms, we get:

-3x - 2 < 10

Adding 8 to both sides, we get.

-3x < 12

Dividing both sides by -3, we get:

x > -4

Combining the solutions from both cases, we get:

x < 2.67 or x > -4

Inequality 2:

1.5x-3<|x- 1|

To solve this inequality, we can use the following steps:

Split the inequality into two cases, one for when the absolute value is positive and one for when the absolute value is negative.

Solve each case individually.

Combine the solutions from each case to get the final solution.

Case 1:

1.5x-3 < x - 1

Subtracting x from both sides, we get:

0.5x - 3 < -1

Adding 3 to both sides, we get:

0.5x < 2

Dividing both sides by 0.5, we get:

`\sf (0.5x)/(0.5) <( 2)/(0.5)`

x < 4

Case 2: 1.5x-3 < -(x - 1)

Distributing the negative sign, we get:

1.5x - 3 > -x + 1

Adding x to both sides, we get:

2.5x - 3 > 1

Adding 3 to both sides, we get

2.5x > 4

Dividing both sides by 2.5, we get:

`\sf (2.5x )/(2.5) >( 4)/(2.5)`

x > 1.6

Combining the solutions from both cases, we get:

x < 4 or x > 1.6

Therefore, the solution to the system of inequalities is:

x < 2.67 or x > -18 and x < 4 or x > 1.6