14.1: Solutions to Equations and Solutions to Inequalities

Solve $\textx = 10$

Find 2 solutions to $\textx > 10$

Solve $2x = \text20$

Find 2 solutions to $2x > \text20$
Let’s solve more complicated inequalities.
Solve $\textx = 10$
Find 2 solutions to $\textx > 10$
Solve $2x = \text20$
Find 2 solutions to $2x > \text20$
Andre has a summer job selling magazine subscriptions. He earns \$25 per week plus \$3 for every subscription he sells. Andre hopes to make at least enough money this week to buy a new pair of soccer cleats.
What are some other numbers of magazine subscriptions Andre could have sold and still reached his goal?
Write an inequality to describe the number of subscriptions Andre must sell to reach his goal.
Diego has budgeted \$35 from his summer job earnings to buy shorts and socks for soccer. He needs 5 pairs of socks and a pair of shorts. The socks cost different amounts in different stores. The shorts he wants cost \$19.95.
Write an inequality to represent the amount Diego can spend on a single pair of socks.
We could express all the values that would work using either \(x \leq \text{__ or } x \geq \text{__}\). Which one should we use?
A teacher wants to buy 9 boxes of granola bars for a school trip. Each box usually costs \$7, but many grocery stores are having a sale on granola bars this week. Different stores are selling boxes of granola bars at different discounts.
Jada and Diego baked a large batch of cookies.
After all this, they had 15 cookies left. How many cookies did they bake?
Suppose Elena has \$5 and sells pens for \$1.50 each. Her goal is to save \$20. We could solve the equation $1.5x + 5 = 20$ to find the number of pens, $x$, that Elena needs to sell in order to save exactly \$20. Adding 5 to both sides of the equation gives us $1.5x = 15$, and then dividing both sides by $1.5$ gives the solution $x=10$ pens.
What if Elena wants to have some money left over? The inequality $1.5x + 5 > 20$ tells us that the amount of money Elena makes needs to be greater than \$20. The solution to the previous equation will help us understand what the solutions to the inequality will be. We know that if she sells 10 pens, she will make \$20. Since each pen gives her more money, she needs to sell more than 10 pens to make more than \$20. So the solution to the inequality is $x > 10$.