Learn on PengiReveal Math, AcceleratedUnit 8: Solve Problems Using Equations and Inequalities

Lesson 8-7: Write and Solve Two-Step Inequalities

In Lesson 8-7 of Reveal Math Accelerated, Grade 7 students learn to write and solve two-step inequalities of the form px + q > r using the Properties of Inequality, including reversing the inequality symbol when multiplying or dividing by a negative number. The lesson covers defining variables, applying inverse operations, and graphing solution sets on a number line. Real-world contexts such as raffle ticket sales and wildlife research are used to practice translating word problems into two-step inequalities.

Section 1

Translate Words to an Inequality

Property

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some common phrases are 'is greater than' ( $>$ ), 'is at least' ( $\geq$ ), 'is less than' ( $<$ ), and 'is at most' ( $\leq$ ).

Examples

'Thirty less than a number $n$ is at least 50' translates to $n - 30 \geq 50$ .
'Four times a number $y$ is no more than 24' translates to $4y \leq 24$ .
'A number $p$ increased by 10 exceeds 25' translates to $p + 10 > 25$ .

Explanation

Math has its own language. Learning keywords helps you translate from English to an inequality. 'Exceeds' means 'greater than,' while 'at most' means 'less than or equal to.' Pay close attention to these phrases.

Section 2

Solving Two-Step Inequalities

Property

Solve an inequality just like an equation by using inverse operations to isolate the variable. Whatever you do to one side, you must do to the other to keep it balanced. For example, to solve $3x + 1 \le 7$ , you first subtract 1 from both sides, then divide both sides by 3.

Examples

$4x + 3 \le 11 \rightarrow 4x \le 8 \rightarrow x \le 2$
$2y - 5 > 9 \rightarrow 2y > 14 \rightarrow y > 7$
$8 \le x - 3 \rightarrow 11 \le x$ , which is the same as $x \ge 11$

Explanation

Solving an inequality is like being a secret agent on a mission to isolate the variable 'x'! First, you have to get rid of any sidekicks hanging around by adding or subtracting. Then, if 'x' has a coefficient partner, you divide to uncover its true identity. Just follow the steps to crack the code!

Section 3

Application: Solving Real-World Problems with Multi-Step Inequalities

Property

A multi-step inequality models real-world scenarios involving a fixed cost plus a variable rate, or comparing two different plans.

Define a variable (e.g., let $x$ be the number of items).
Translate keywords: "at least" ( $\geq$ ), "at most" ( $\leq$ ), "more than" ( $>$ ), "fewer than" ( $<$ ).
Build the inequality: Fixed Amount + (Rate * Variable).
Solve and interpret the result practically (e.g., you cannot buy half a ticket).

Examples

Example 1 (Budget Constraint): A gym membership costs $30 per month plus$ 5 per guest pass. If you want to spend at most $55 this month, how many guest passes ($ g$) can you buy?

Inequality: $30 + 5g \leq 55$ .
Solve: Subtract 30 to get $5g \leq 25$ , then divide by 5 to get $g \leq 5$ . You can buy at most 5 guest passes.

Example 2 (Comparing Plans): Plan A charges a $15 monthly fee plus$ 0.10 per text. Plan B charges $0.25 per text with no fee. For how many texts ($ t$) is Plan A cheaper (costs less than Plan B)?

Inequality: $15 + 0.10t < 0.25t$ .
Solve: Subtract $0.10t$ from both sides to get $15 < 0.15t$ . Divide by 0.15 to get $100 < t$ (which is $t > 100$ ). Plan A is cheaper if you send more than 100 texts.

Example 3 (Discrete Limits): A phone plan costs $40 per month plus$ 0.15 per text. To keep your bill strictly under $50, how many texts ($ t$) can you send?

Inequality: $40 + 0.15t < 50$ .
Solve: $0.15t < 10 \rightarrow t < 66.67$ . Since you cannot send a fraction of a text, you can send at most 66 texts.

Explanation

Real-world math rarely requires just one step! Most scenarios involve a starting fee that happens once, plus a rate that happens repeatedly. When translating these into math, place your variable next to the rate. Multi-step inequalities are incredibly powerful for making financial decisions, like figuring out exactly when a subscription plan with an upfront fee becomes a better deal than a pay-as-you-go plan.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Translate Words to an Inequality

Property

Examples

'Thirty less than a number $n$ is at least 50' translates to $n - 30 \geq 50$ .
'Four times a number $y$ is no more than 24' translates to $4y \leq 24$ .
'A number $p$ increased by 10 exceeds 25' translates to $p + 10 > 25$ .

Explanation

Section 2

Solving Two-Step Inequalities

Property

Examples

$4x + 3 \le 11 \rightarrow 4x \le 8 \rightarrow x \le 2$
$2y - 5 > 9 \rightarrow 2y > 14 \rightarrow y > 7$
$8 \le x - 3 \rightarrow 11 \le x$ , which is the same as $x \ge 11$

Explanation

Section 3

Application: Solving Real-World Problems with Multi-Step Inequalities

Property

A multi-step inequality models real-world scenarios involving a fixed cost plus a variable rate, or comparing two different plans.

Define a variable (e.g., let $x$ be the number of items).
Translate keywords: "at least" ( $\geq$ ), "at most" ( $\leq$ ), "more than" ( $>$ ), "fewer than" ( $<$ ).
Build the inequality: Fixed Amount + (Rate * Variable).
Solve and interpret the result practically (e.g., you cannot buy half a ticket).

Examples

Example 1 (Budget Constraint): A gym membership costs $30 per month plus$ 5 per guest pass. If you want to spend at most $55 this month, how many guest passes ($ g$) can you buy?

Inequality: $30 + 5g \leq 55$ .
Solve: Subtract 30 to get $5g \leq 25$ , then divide by 5 to get $g \leq 5$ . You can buy at most 5 guest passes.

Example 2 (Comparing Plans): Plan A charges a $15 monthly fee plus$ 0.10 per text. Plan B charges $0.25 per text with no fee. For how many texts ($ t$) is Plan A cheaper (costs less than Plan B)?

Example 3 (Discrete Limits): A phone plan costs $40 per month plus$ 0.15 per text. To keep your bill strictly under $50, how many texts ($ t$) can you send?

Inequality: $40 + 0.15t < 50$ .
Solve: $0.15t < 10 \rightarrow t < 66.67$ . Since you cannot send a fraction of a text, you can send at most 66 texts.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Translate Words to an Inequality

Property

Examples

'Thirty less than a number $n$ is at least 50' translates to $n - 30 \geq 50$ .
'Four times a number $y$ is no more than 24' translates to $4y \leq 24$ .
'A number $p$ increased by 10 exceeds 25' translates to $p + 10 > 25$ .

Explanation

Section 2

Solving Two-Step Inequalities

Property

Examples

$4x + 3 \le 11 \rightarrow 4x \le 8 \rightarrow x \le 2$
$2y - 5 > 9 \rightarrow 2y > 14 \rightarrow y > 7$
$8 \le x - 3 \rightarrow 11 \le x$ , which is the same as $x \ge 11$

Explanation

Section 3

Application: Solving Real-World Problems with Multi-Step Inequalities

Property

A multi-step inequality models real-world scenarios involving a fixed cost plus a variable rate, or comparing two different plans.

Define a variable (e.g., let $x$ be the number of items).
Translate keywords: "at least" ( $\geq$ ), "at most" ( $\leq$ ), "more than" ( $>$ ), "fewer than" ( $<$ ).
Build the inequality: Fixed Amount + (Rate * Variable).
Solve and interpret the result practically (e.g., you cannot buy half a ticket).

Examples

Example 1 (Budget Constraint): A gym membership costs $30 per month plus$ 5 per guest pass. If you want to spend at most $55 this month, how many guest passes ($ g$) can you buy?

Inequality: $30 + 5g \leq 55$ .
Solve: Subtract 30 to get $5g \leq 25$ , then divide by 5 to get $g \leq 5$ . You can buy at most 5 guest passes.

Example 2 (Comparing Plans): Plan A charges a $15 monthly fee plus$ 0.10 per text. Plan B charges $0.25 per text with no fee. For how many texts ($ t$) is Plan A cheaper (costs less than Plan B)?

Example 3 (Discrete Limits): A phone plan costs $40 per month plus$ 0.15 per text. To keep your bill strictly under $50, how many texts ($ t$) can you send?

Inequality: $40 + 0.15t < 50$ .
Solve: $0.15t < 10 \rightarrow t < 66.67$ . Since you cannot send a fraction of a text, you can send at most 66 texts.