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How do you solve the inequality \[4x - 2 < 6\] or \[3x + 1 > 22\]?

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Last updated date: 25th Jun 2024
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Answer
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Hint: An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. Here we need to solve for ‘x’ which is a variable. Solving the given inequality is very like solving equations and we do most of the same thing but we must pay attention to the direction of inequality\[( \leqslant , > )\]. We have two linear inequalities. We can solve this.

Complete step-by-step solution:
Given, \[4x - 2 < 6\]
We add 2 on both sides and we know that the inequality direction doesn’t change,
 \[
  4x - 2 + 2 < 6 + 2 \\
  4x < 8 \\
 \]
We divide 4 on both sides we have,
\[
  x < \dfrac{8}{4} \\
  x < 2 \\
 \]
Thus the solution of \[4x - 2 < 6\] is \[x < 2\]. The interval form is \[( - \infty ,2)\].
Now take \[3x + 1 > 22\]
Subtract 1 on both sides we have,
\[
  3x + 1 - 1 > 22 - 1 \\
  3x > 21 \\
 \]
Divide by 3 on both sides we have,
\[
  x > \dfrac{{21}}{3} \\
  x > 7 \\
\]
Thus the solution of \[3x + 1 > 22\] is \[x > 7\]. The interval form is \[(7,\infty )\].

Note: We take value of ‘x’ in \[( - \infty ,2)\] and put it in \[4x - 2 < 6\]
Let’s put \[x = 0\] in \[4x - 2 < 6\]
\[ 4(0) - 2 < 6 \\
   - 2 < 6 \\
\]
Which is correct. We check for the second inequality in the same way.
We know that \[a \ne b\] says that ‘a’ is not equal to ‘b’. \[a > b\] means that ‘a’ is less than ‘b’. \[a < b\] means that ‘a’ is greater than ‘b’. These two are known as strict inequality. \[a \geqslant b\] means that ‘a’ is less than or equal to ‘b’. \[a \leqslant b\] means that ‘a’ is greater than or equal to ‘b’.

The direction of inequality do not change in these cases:
-Add or subtract a number from both sides.
-Multiply or divide both sides by a positive number.
-Simplify a side.

The direction of the inequality change in these cases:
-Multiply or divide both sides by a negative number.
-Swapping left and right hand sides.