How to solve Linear equalities and linear inequalities

Linear equation

Example 1
Solve for x: 3 x – 7 > 20.

equation

To check the solution, first see whether x = 9 makes the equation 3 x – 7 = 20 true. Even though 9 isn't a solution, it's a critical number or dividing point and is important to finding the solution.

equation

Now, choose a number greater than 9—10, for example, and see whether that makes the original inequality true.

equation

This is a true statement. Since it is impossible to list all the numbers that are greater than 9, use “set builder” notation to show the solution set.

{ x| x > 9}

This is read as “the set of all x so that x is greater than 9.” Many times, the solutions to inequalities are graphed to illustrate the answers. The graph of { x| x > 9}is shown in Figure 1.

Figure 1. Note that 9 is not included.
figure

Example 2
Solve for x: equation.

The LCD for the denominators in this inequality is 24. Multiply both sides of the inequality by 24 as you would have had this been an equation.

equation

At this point, you can isolate x on either side of the inequality.

equation

In the final step on the left, the direction is switched because both sides are multiplied by a negative number. Both methods produce the final result that says that x is a number less than equation. The check is left to you. The solution set is expressed as

equation

The graph of this solution set is shown in Figure 2.

Figure 2. Note the hole at equation.
figure

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