Linear Inequalities - Examples and Practice Problems

• We start by writing the original problem:

$latex 3x-5>1$

• To solve for the variable, we add 5 to both sides of the inequality:

$latex 3x-5+5>1+5$

• After simplifying, the expression reduces to:

$latex 3x>6$

• To solve, we divide both sides by 3:

$latex \frac{3}{3}x> \frac{6}{3}$

$latex x> 2$

• We graph the inequality with an open point since 2 is not included in the solution. The solution is all the numbers to the right of 2:

Step 1: Here, we have nothing to simplify, so we start with:

$latex 5x-10<15$

Step 2: To solve for the variable, we add 10 from both sides and simplify:

$latex 5x-10+10<15+10$

$latex 5x<25$

Step 3: To solve, we divide both sides by 5:

$latex \frac{5}{5}x<\frac{25}{5}$

$latex x<5$

Step 4: To graph, we note that the solutions to the inequality are all real numbers to the left of 5. The 5 is not included, so we use an empty point to indicate this:

Step 1: We have nothing to simplify, so we start with:

$latex -4x-5\leq 3$

Step 2: We add 5 to both sides to solve for the variable:

$latex -4x-5+5\leq 3+5$

$latex -4x\leq 8$

Step 3: We divide both sides by -4 to get:

$latex \frac{-4}{-4}x\leq\frac{8}{-4}$

$latex x\geq -2$

Step 4: In this case, -2 is part of the solution. Therefore, we use a solid point to indicate that the solutions are all numbers to the right of -2, including -2:

In this case, we have variables on both sides. We have to move the variables to one side and the constants to the other. It doesn’t matter which side contains the variables, but it is common to move the variables to the left:

• We start with the original problem:

$latex 4x+2\geq 2x+10$

• We subtract 2 and 2x from both sides to solve for the variable:

$$4x+2-2-2x\geq 2x+10-2-2x$$

• Simplifying the inequality, we have:

$latex 2x\geq 8$

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

$latex \frac{2}{2}x\geq \frac{8}{2}$

$latex x\geq 4$

• Here, the 4 is part of the solution, so we use a closed point to indicate this:

Step 1: We have nothing to simplify. We start with the inequality:

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

Step 2: We subtract 3 and 3x from both sides to solve for the variable:

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

$latex 2x>-6$

Step 3: We divide both sides by 2 to solve:

$latex \frac{2}{2}x>\frac{-6}{2}$

$latex x>-3$

Step 1: We have parentheses, so we apply the distributive property to eliminate them:

$latex 3(x+2)>-9$

$latex 3x+6>-9$

Step 2: To solve for the variable, we subtract 6 from both sides:

$latex 3x+6-6>-9-6$

$latex 3x>-15$

Step 3: To solve, we divide both sides by 3:

$latex \frac{3}{3}x>\frac{-15}{3}$

$latex x>-5$

In this case, we have parentheses, so we use the distributive property to remove parentheses and simplify:

• We write the original problem:

$latex 2(3x-3)>4x$

• We apply the distributive property:

$latex 2(3x)+2(-3)>4x$

$latex 6x-6>4x$

• We add 6 from both sides and subtract 4x to solve for the variable:

$$6x-6+6-4x>4x+6-4x$$

• After simplifying, the expression reduces to:

$latex 2x>6$

• By dividing both sides by 2, we have:

$latex \frac{2}{2}x> \frac{6}{2}$

$latex x> 3$

Step 1: We simplify the parentheses and combine like terms:

$latex 2(2x+4)+5>1$

$latex 4x+8+5>1$

$latex 4x+13>1$

Step 2: We isolate the variable by subtracting 13 from both sides:

$latex 4x+13-13>1-13$

$latex 4x>-12$

Step 3: We have to divide by 4:

$latex \frac{4}{4}x>\frac{-12}{4}$

$latex x>-3$

Step 1: We simplify the parentheses on both sides and combine like terms:

$latex 4(2x+4)-3\leq 2(3x+4)+3$

$latex 8x+16-3\leq 6x+8+3$

$latex 8x+13\leq 6x+11$

Step 2: We subtract 13 and 6x from both sides to solve for the variable:

$$8x+13-13-6x\leq 6x+11-13-6x$$

$latex 2x\leq -2$

Step 3: To solve, we divide both sides by 2:

$latex \frac{2}{2}x\leq \frac{-2}{2}$

$latex x\leq -1$

Step 1: We simplify the parentheses on both sides and combine like terms:

$latex 2(x+5)-10\geq 4(2x+6)$

$latex 2x+10-10\geq 8x+24$

$latex 2x\geq 8x+24$

Step 2: We subtract both sides by 8x to solve for x:

$latex 2x-8x\geq 8x+24-8x$

$latex -6x\geq 24$

Step 3: Now, we divide by -6:

$latex \frac{-6}{-6}x\geq\frac{24}{-6}$

$latex x\leq -4$