Solve the following compound inequality. Write the solution set using interval notation and graph it. $$ 1-xx $$

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Solve the following compound inequality. Write the solution set using interval notation and graph it. $$ 1-x<7+x \text { or } 6 x+5>x $$

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Solve the inequality by following steps: - step0: Solve for $$x$$: $$6x+5>x$$ - step1: Move the variable to the left side: $$6x+5-x>0$$ - step2: Subtract the terms: $$5x+5>0$$ - step3: Move the constant to the right side: $$5x>0-5$$ - step4: Remove 0: $$5x>-5$$ - step5: Divide both sides: $$\frac{5x}{5}>\frac{-5}{5}$$ - step6: Divide the numbers: $$x>-1$$ Solve the equation $$ 1-x<7+x $$. Solve the inequality by following steps: - step0: Solve for $$x$$: $$1-x<7+x$$ - step1: Move the expression to the left side: $$-x-x<7-1$$ - step2: Add and subtract: $$-2x<7-1$$ - step3: Add and subtract: $$-2x<6$$ - step4: Change the signs: $$2x>-6$$ - step5: Divide both sides: $$\frac{2x}{2}>\frac{-6}{2}$$ - step6: Divide the numbers: $$x>-3$$ To solve the compound inequality $$ 1 - x < 7 + x \text{ or } 6x + 5 > x, $$ we will solve each part separately. ### Step 1: Solve the first inequality $$1 - x < 7 + x$$ 1. Rearranging the inequality: $$ 1 - x < 7 + x $$ Add $$x$$ to both sides: $$ 1 < 7 + 2x $$ Subtract $$7$$ from both sides: $$ -6 < 2x $$ Divide by $$2$$: $$ -3 < x \quad \text{or} \quad x > -3. $$ ### Step 2: Solve the second inequality $$6x + 5 > x$$ 1. Rearranging the inequality: $$ 6x + 5 > x $$ Subtract $$x$$ from both sides: $$ 5 > -5x $$ Divide by $$-5$$ (remember to flip the inequality sign): $$ -1 < x \quad \text{or} \quad x > -1. $$ ### Step 3: Combine the results The solution sets from the inequalities are: - From the first inequality: $$x > -3$$ - From the second inequality: $$x > -1$$ Since this is an "or" compound inequality, we take the union of the two solution sets. The more restrictive condition is $$x > -1$$. ### Final Solution The solution set in interval notation is: $$ (-1, \infty). $$ ### Graphing the Solution To graph the solution: - Draw a number line. - Place an open circle at $$-1$$ (indicating that $$-1$$ is not included). - Shade the line to the right of $$-1$$ to indicate all numbers greater than $$-1$$. This represents the solution set $$ (-1, \infty) $$. The solution set is $$ x > -1 $$, which in interval notation is $$ (-1, \infty) $$.

Solve the following compound inequality. Write the solution set using interval notation and graph it.

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