How to Solve Linear Inequalities?

To solve the inequality, we want to find all the x values that satisfy it. This means that there are an infinite number of solutions that, when substituted, would make the inequality true. Therefore, we follow the following steps:

• We write the original problem:

$latex 4x-6>2$

• We add 6 to both sides to keep the variables on one side and the constants on the other:

$latex 4x-6+6>2+6$

• After simplifying, the expression reduces to:

$latex 4x>8$

• We divide both sides by 4 and simplify to get the answer:

$latex \frac{4}{4}x>\frac{8}{4}$

$latex x>2$

• We use an open point to indicate that 2 is not part of the solution. The solution to the inequality includes all values to the right of 2, but does not include 2:

This is an example of what happens to the inequality when we divide by a negative number:

• We write the original problem:

$latex -2x-3\geq 1$

• We isolate the variable by adding 3 to both sides:

$latex -2x-3+3\geq 1+3$

• After simplifying, the expression reduces to:

$latex -2x\geq 4$

• We divide both sides by -2 and simplify to get the answer:

$latex \frac{-2}{-2}x\geq \frac{4}{-2}$

$latex x\leq -2$

• We use a closed point to indicate that the -2 is part of the solution. The solution to the inequality includes -2 and all values to the left:

Here, we have variables on both sides. It is possible to place the variables on the left side or on the right side, but the standard is to do it on the left side.

• We write the original problem:

$latex 5-3x>13-5x$

• We can add 5x to both sides and subtract 5 to solve for the variable:

$latex 5-3x+5x-5>13-5x+5x-5$

• Simplifying the expression, we have:

$latex 2x>8$

• We can get the answer by dividing both sides by 2:

$latex \frac{2}{2}x> \frac{8}{2}$

$latex x> 2$

In this case, we have parentheses, so we have to start by applying the distributive property to remove the parentheses:

• We write the original problem:

$latex 2(2x+3)<2x+12$

• We apply the distributive property:

$latex 2(2x)+2(3)<2x+12$

$latex 6x+6<2x+12$

• We isolate the variable by subtracting 6 and 2x from both sides:

$latex 6x+6-2x-6<2x+12-2x-6$

• After simplifying, the expression reduces to:

$latex 4x<8$

• We divide both sides by 4 and simplify to get the answer:

$latex \frac{4}{4}x< \frac{8}{4}$

$latex x< 2$

Again, we start by applying the distributive property to remove the parentheses:

• We write the original problem:

$latex 2(3x-4)\geq 7x-7$

• We apply the distributive property:

$latex 2(3x)+2(-4)\geq 7x-7$

$latex 6x-8\geq 7x-7$

• Now, we add 8 and subtract 7x from both sides to isolate the variable:

$latex 6x-8+8-7x\geq 7x-7+8-7x$

• We simplify to obtain:

$latex -x\geq 1$

• We divide both sides by -1 and simplify to get the answer:

$latex \frac{-1}{-1}x\geq \frac{1}{-1}$

$latex x\geq -1$

In this case, we have to apply the distributive property to both parentheses and then combine like terms to simplify:

• We write the original problem:

$latex 3(x+2)+2\leq 2(x-1)+8$

• We apply the distributive property to both parentheses:

$latex 3(x)+3(2)+2\leq 2(x)+2(-1)+8$

$latex 3x+6+2\leq 2x-2+8$

• We combine like terms to simplify:

$latex 3x+8\leq 2x+6$

• We isolate the variable by subtracting 8 and 2x from both sides:

$latex 3x+8-8-2x\leq 2x+6-8-2x$

• Simplifying, we have:

$latex x\leq -2$

• We no longer have to divide:

$latex x\leq -2$